2.1. Novel thalamocortical cell model is necessary to capture experimentally observed awake dynamics

Experimental studies (discussed in detail in the Methods) show that by increasing depolarization via an applied current, in the presence of a metabotropic glutamate receptor (mGluR1) agonist, a thalamocortical cell produces five distinct dynamic firing types [21,28,29]. These studies also reveal that at least one of these novel firing types, the high-threshold bursting alpha rhythm, is exhibited in the presence of cholinergic (mACHr) agonists [30]. Since previous models failed to capture the full observed dynamic range of these thalamocortical cells using validated ion channels (discussed in Methods), we constructed a new cellular model that contains both an M current () and a high-threshold L-type calcium current (). Notably, all currents used in our model are expressed within the thalamus [31–33] and the ranges for their associated parameter values are restricted based on current clamp recordings in manners (and values) consistent with previous modeling studies (see Methods for details) [18,31,34,35].

Our model produces five distinct dynamic firing types, including the elusive experimentally observed high-threshold alpha-frequency bursting state [30,36]. The full dynamic range in the model can be found by decreasing the background potassium conductances, an action used in computational modeling to mimic either an applied cholinergic or glutamatergic agonist effect [18,31,35], and then continually increasing in order to depolarize the cell (Figs 1 and S1). Additional key pharmacological blocking experiments are replicated within the supplement [30,36,37] to show further relationships between the different ionic conductances and the dynamic firing regimes of the model (S2 Fig).

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Fig 1. Thalamocortical cell model produces five distinct complex firing dynamics.

With fixed ionic conductance values and increasing applied current () from left to right, the model produces some notable firing patterns highlighted in the experimental work of Crunelli, Lorincz, and Hughes [27,28]. In order, the model dynamics are: I and II low-threshold bursting, III slow tonic firing, IV high-threshold bursting and V fast tonic spiking. A more detailed depiction (for 12 fixed values of ) of the possible neuron dynamics is presented in (S1 Fig). Note, the distinction between low-threshold bursting dynamic states I and II are laid out in the next subsection. For comparison with experimental literature see: Fig 1 in [28] and Figs 1 and 2 in [37].

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Critically, in addition to resembling experimentally observed firing types, these states can also be related to levels of arousal [18,27,28,38]. Fig 1I and 1II show two low-threshold bursting states. The range of frequencies of low-threshold bursting includes <1Hz frequencies that correspond to the Slow-wave sleep rhythms and between Hz frequencies that aligns with observed delta (1–4Hz) sleep rhythms (S1 Fig) [38]. The model also suggests that low-threshold bursting at greater than delta frequencies is possible (S1 Fig), which is discussed in greater detail in the next subsection in the context of the experimental literature. In addition to the sleep states, the model produces three distinct awake firing dynamics (Fig 1). Note that these states are categorized as “awake” firing based on minimum observed membrane potentials (mV, discussed in Methods) and the lack of hyperpolarization events [27,29,37]. The first awake related firing type is a slow tonic spiking mode (missing from previous models [28]) that happens at theta (4–8Hz) to alpha frequencies (8–14Hz) (Figs 1 and S1). With slightly more applied current (), in Figs 1IV and S1 the model generates high-threshold bursts at theta and alpha frequencies, with experimentally matching intra-burst frequencies [27,30,36] (see section 2.9 for more details). Finally, additional increases in results in a fast tonic firing state at frequencies beta (20Hz) and above. The first two awake states (Fig 1III and 1IV) correspond to an experimental awake behavioral state termed "natural wakefulness" in which prominent alpha rhythms can rapidly emerge in occipital EEG by closing the eyes [21,29]. The fast tonic firing seen in Fig 1V, is the most excited state, with a firing frequency that is associated with attentional processing [18,39,40].

While the five states in Fig 1 were found under fixed ionic conductance levels by varying the external excitation , experimental work has shown some of these state transitions occur in the presence of neuromodulation [41–44]. Specifically, glutamate (Glu) and acetylcholine (ACh) are implicated in the transition from sleep to waking states via their respective activation of metabotropic glutamate receptors (mGluRs) and muscarinic acetylcholine receptors (mAChRs) [42,44–48]. The next subsection investigates how these neuromodulatory signals act in the model, both independently and in tandem with changes in external excitation.

2.2. Neuromodulatory arousal signals switch the thalamic cell from low threshold bursting to awake dynamic states

It is well documented that glutamatergic signals from the cortex and cholinergic signals from the brainstem play a strong role in transitioning the thalamus from sleep states to wakeful states [42,44–48]. These arousal signals are typically instantiated in conductance-based thalamic models by decreasing the background potassium channel conductances of or (see Methods for ionic current details) [18,35,43,49]. This action replicates one (of many) downstream effects of glutamate and acetylcholine via distinct signaling pathways involving mGluR and mAChR receptor (both expressed throughout the thalamus) activation [32,33,44,50–53]. Depending on receptor and ion channel expression levels these two neuromodulatory signals may alter one, or both, of or to varying degrees; for simplicity, moving forward we refer not to the specific neuromodulator but to the modified ionic current. In this subsection, the M-current conductance () is fixed and the potassium leak conductance () is varied. In a later subsection the M-current conductance is varied while is fixed, showing that has a more nuanced impact on the observable dynamics. The differences between glutamatergic and cholinergic arousal signaling (and their impacts on modeling) are examined in greater detail in the discussion section 3.2.

Fig 2 shows how model arousal state varies with changes in potassium leak conductance and applied external current. The center diagram in Fig 2 plots the dynamic behavior of the TC cell as a function of (x-axis) and (y-axis). Each solid curve (black, purple) denotes a transition in parameter space from one dynamic firing state to another. The purple curve is highlighted as a key transition from sleep to wakefulness (STW). Each sectioned region (0-V) is categorized by their exemplified firing dynamics (i.e., quiescent, low-threshold bursting, high-threshold bursting. tonic spiking, or some combination of these) provided in the accompanying voltage plots. Dynamic state labels are consistent with Fig 1.

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Fig 2. Description of the relationship between the thalamocortical model state of arousal, potassium leak conductance and external excitation ().

Solid lines in the central plot are used to divide parameter space into a total of six dynamically distinct regions (0-V). The magenta curve denotes the sleep-to-wake (STW) transition. Dashed orange and blue lines show transitions through parameter space either vertically or horizontally by fixing the corresponding parameter. Sample voltage traces for the TC cell from within each region are provided. The order of firing dynamics for these regions corresponds to the dynamics of Fig 1. Two differences from Fig 1: the quiescent hyperpolarized state (0) is a newly described state, and regions II and III have two different corresponding membrane potential plots (depicted in red dashed boxes) depending on the cell’s initial conditions. Note the complete bifurcation diagram with labeled mathematical bifurcation curves can be found in S3 Fig.

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Fig 2 also reveals additional dynamic features not captured by Fig 1, namely that regions II and III are bistable; different initial conditions can lead to two different firing type outputs. Importantly, one of the bistable states in region II is a depolarized quiescent state that is commonly observed during slice experiments when transitioning from low-threshold bursting to tonic spiking with increasing [28]. This bistability (and existence of the depolarized quiescent state) of the model is critical for explaining why frequencies higher than delta (1–4Hz) can be produced by our model even though they are not typically seen in-vitro. The model shows that experimentally observed low-threshold bursting frequencies (between 0–6Hz) occur in region I as the only stable state, while the higher (not typically experimentally observed) frequencies occur only in the bistable region II (S4 Fig). However, in the absence of external inputs that can activate the T-type calcium channel, the model suggests that low-threshold bursts at frequencies greater than delta will be rare events: instead the silent depolarized state will emerge as is commonly seen with increasing in experiments [28].

Experimentally, the full dynamic range was observed in slice experiments only after application of a glutamatergic agonist [29,30]. Without glutamatergic agonist application, experimentalists observed a smaller number of distinct dynamic states [28]. The model could help explain this experimental observation with the central plot of Fig 2 by considering two levels of modulated (proposed conductance altered by the applied agonist) and varying across an identical fixed range (blue horizontal dashed lines). At high (low levels of agonist), the model reaches only three states whereas all 5 states are reached with low (high levels of agonist) for an identical range of (Fig 2). Furthermore, the bistability of region II (combined with the increased width of region II for high ) could help explain why, prior to agonist application, experimentalists could observe a quiescent state that was nonexistent (model quickly transitions from state I to state III for low ) afterwards [28]. These results indicate that neuromodulatory changes of could increase the total number of possible observed dynamic states, while also impacting which state transitions are possible.

Fig 2 also shows the impact of modulating if external excitation is kept constant (orange vertical dashed lines). Starting in any of the awake states (III,IV,V), for any fixed value of , increasing from 0 brings the neuron to a lower state of arousal. In spite of this, one subtle observation is that, with fixed excitation, the neuron cannot transition through the full range of states purely with manipulation (Fig 2 orange lines). While modulation of is sufficient to transition the thalamus between light sleep and resting awake states, reaching the slowest low-threshold bursting frequencies (<1Hz) associated with Slow-wave sleep (found at the left edge of region I in the model), from any of the awake states (III,IV,V), requires additional changes in external thalamocortical cell input (S4 Fig) [42].

This model is uniquely positioned to explore how retinogeniculate transmission changes with arousal. It is crucial that the model captures the full dynamic range in order to identify differences in transmission during sleep states (Fig 1I and 1II), the natural wakefulness states (Fig 1III and 1IV), and the highly excited state (Fig 1V). How transmission probability changes as a function of LGN state is the focus of the next subsections.

2.3. LGN transmission of retinal signals depends on model arousal level

It is well known the retina can send spiking signals to the LGN during each of the different states of arousal [54,55]. To investigate how these signals are processed as arousal level is changed, a spiking retinal ganglion cell is connected via an excitatory AMPA synapse to the model LGN thalamocortical cell (see Methods). The interspike intervals for the simulated retinal spike trains are then generated from a gamma distribution (see Methods) [10,56]. Fig 3 shows how retinal input is processed by the LGN during the different LGN arousal states.

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Fig 3. LGN response to retinal input changes with arousal state.

The five arousal states match the order and meaning from Fig 1. Each plot shows how LGN spiking (in the 5 different arousal states) responds to an identical retinal input that is turned on when time reaches 2000ms. Retinal spikes occur at time points denoted by red tick marks. Note, retinal ISIs are drawn from a gamma distribution with a mean of 60 spikes/sec.

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The five distinct model arousal states differ in their processing of visual information from the retina. In states I and II, retinal spikes are not patterning the output but instead act as blanket excitation by increasing the low-threshold bursting frequency. This is confirmed when replacing the retinal inputs with a fixed increase in during states I and II yields a similar thalamocortical cell output (Figs 3II, S5A, S5B). While there is an increase in frequency of the low-threshold bursting, the timing of the retinal spikes is irrelevant for patterning the LGN output as they are not strong enough to transition the model out of the sleep-like states. In contrast, in the natural wakefulness states III and IV (Fig 3) the model enters what we call a filtering mode. The TC cell is responsive to the retinal inputs but only a subset of the retinal spikes are transmitted; as the model progresses from state III to IV there is a continuous increase in retinal transfer. Finally, in the fast tonic firing regime (state V) [18,57], there is a nearly faithful transmission of retinal spikes regardless of the spike timing. This is corroborated by the accompanying delay time and transfer frequency (between retinal and LGN spikes) statistics: I (8.5ms, N/A), II (9.3ms, N/A), III (3.0ms, .4), IV (3.2ms, .63), and V (2.5ms, .99) (Fig 3). Note, transmission percentages for state I and II are given as N/A as the cell remains in low-threshold burst mode with average delay times greater than 5ms (section 2.5 will provide detailed descriptions of the delay time and transfer frequency metrics [9,58]).

There are a couple of nuanced points regarding the response of state II to retinal inputs that result from the bistability of the region and the interactions with the sleep-to-wake border (Fig 2). First, if parameters are set such that the model is in State II but very close to the sleep-to-wake border, small amounts of excitation, such as what is provided by incoming retinal input, may allow the model to transition into State III and thus filter the spiking input (S5C and S5D Fig). Second, we predict that the quiescent state of region II is more likely to be observed (as it is the stable state found after noise cessation) for low neuromodulatory tone (high ) and for levels closer to the sleep-to-wake border (S3B and S6 Figs). This reveals two ideas: first, it provides further evidence for why, in the Crunelli experiments, the quiescent state was observed only prior to agonist application [28]. Second, it suggests that during slice experiments, the use of noisy stimuli could help correctly characterize firing regimes, uncover if there is any type of bistability and decipher stable versus unstable steady states (S6 Fig).

Fig 3 provides individual representations of the LGN response to retinal inputs in the different dynamic regions. The following subsection explores how variable the thalamic outputs can be, particularly during the filtering states III and IV.

2.4. Novel awake dynamic states of the model reproduce experimental variability in mean retinal transmission

In-vivo experimental studies have shown that a given LGN-retinal pair can exhibit transfer frequencies from twenty to one hundred percent [5,7,13]. The source of this variability is unknown. This subsection shows that the novel awake states of the model might be a source of such variability.

By examining how the retinogeniculate filtering statistics change throughout the five thalamic dynamic states, Fig 4 reveals that the vast majority of filtering occurs in the awake states III and IV. Fig 4A is a version of the two parameter plot from Fig 2 that depicts the underlying state dynamics; it is there to compare with Fig 4B and 4C, which convey the retinal transmission statistics. Fig 4B shows the ratio of LGN output spikes to incoming retinal spikes. Fig 4C plots the average delay time between an LGN spike and the nearest preceding retinal input. Fig 4B and 4C together determine whether an area in parameter space is filtering or not. To be in this filtering mode requires two conditions based on experimental findings: 1) the ratio of output LGN spikes to input retinal spikes is less than one, and 2) the average delay time between the LGN spike and retinal spike should be under five milliseconds, suggesting a causal relationship between retinal input and LGN spiking [9,58]. This overlap is depicted in Fig 4D, confirming that filtering is primarily restricted to states III and IV. Fig 4 also confirms that retinal spikes are not patterning thalamic output in states I and II (delay time >5ms) and that all retinal inputs are faithfully transmitted in state V (Transfer Ratio and delay time <5ms). Note that there is a small region of filtering to the left of the sleep-to-wake curve (dashed purple line). This is consistent with the results of the previous subsection showing that retinal input near the sleep-to-wake border can push the model into a filtering state (S5 Fig).

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Fig 4. The filtering of retinal signals coincides in parameter space with the novel awake dynamic regimes and captures observed heterogenity in transmission percentages.

A) Reproduced diagram from Fig 2 for visual comparison with the other panels. B) Transfer probability (colorbar value) between incoming retinal spikes and outgoing LGN neuron spikes plotted as a function of and . C) The delay time in milliseconds (colorbar value) between LGN spikes and the nearest driving incoming retinal spike plotted for the same and parameter space. D) Overlap between B and C where filtering occurs: transfer probability <1 and delay time <5ms. The details for calculating transfer frequency and the delay time are provided in the Methods. The state transition curves (dashed black and purple lines) have been redrawn in Fig 4B-4D for further clarity. The white dashed line in Fig 4D denotes where transfer frequency in the filtering mode reaches .75. For experimental comparison with mean retinal transfer efficacy in resting awake animals see Fig 7 of [5].

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Of note, our model also reproduces the experimentally observed distributions of retinal transfer ratios [5,7,13,15]. Numerous experimental studies have shown that the average overall transfer efficacy rate is near fifty percent with typical values ranging from twenty to seventy percent [5,15]. Fig 4D shows that the model produces a similar distribution of transfer ratios: the majority of parameter space (area between the purple and white dashed lines) where filtering occurs is dedicated to transfer frequencies between .25 and .75. Only for the small area of parameter space to the right of the white line in Fig 4D do transfer ratios reach above .75, indicating that observing these high transmission ratios should be rare.

This subsection shows that the model compares well with experimentally observed mean retinal transfer percentages. The next subsections will examine two more granular details of retinogeniculate transmission: what percentage of LGN spikes are driven by the primary retinal ganglion cell (RGC) and which retinal spikes are preferentially transmitted versus filtered out.

2.5. High-threshold bursting states III and IV preferentially transmit short retinal interspike intervals similarly to experimentally observed awake animal retinal transfer

Retinogeniculate experimental studies have shown that the majority (>90%) of LGN spikes are driven by a primary retinal ganglion cell [9,58]. These studies revealed that the driving RGC spike preceded the LGN spike with a delay time between 0ms and 5ms [9,58]. To compare with these experimental results, Fig 5A plots a histogram of the state IV model delay times between recorded LGN spikes and the immediate closest preceding retinal spike. In agreement with the experimental literature, the majority () of the model delay times are between 0ms and 5ms [9,58]. LGN spikes occurring more than 5ms after the closest retinal spike (), can either be attributed to integration from other non-modeled retinal cells whose inputs are implicitly contained within the excitation term [3,4], or the underlying intrinsic spiking dynamics from state IV. This finding indicates that retinal input largely is sufficient to override the intrinsic model dynamics and dictate the thalamic spike timing.

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Fig 5. LGN spike generation in the awake state is heavily influenced by retinal ISI.

A) Distribution of delay times between each LGN spike and the preceding retinal spike. B) Membrane potential of the LGN TC cell plotted against time. Red tick marks denote timestamps of the incoming retinal spikes. Curly braces depict example retinal ISIs from the three distinct time length categories: short (, 0-10ms), medium (, 10-30ms) and long (, > 30ms). C) Distribution of retinal interspike intervals. The ISIs for the full retinal spike train are plotted in blue. The overlapping orange histogram depicts the ISIs between two retinal spikes where the second spike is an LGN spike driver. Curly braces here depict the three distinct time length categories.

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It known that the successful transmission of RGC spiking inputs is dependent on ISI length [5,6,8,11–13]. Notably, the increased transmission probability for short ISIs compared to long ISIs is more pronounced in anesthetized animals; ISIs greater than 30ms have a near zero percent chance of generating a spike in anesthetized animals as opposed to between chance of producing an LGN response in awake behaving animals [5]. The remaining two panels of Fig 5 examine how retinal spike transmission in the model TC cell during state IV depends on retinal ISI. Fig 5B shows a portion of the voltage trace from the TC cell and the corresponding retinal spike times (red tick marks). Note, Fig 5B labels examples of three distinct retinal ISI categories: short (0–10ms), medium (10–30ms), and long (>30ms). In the accompanying plot of Fig 5C two histograms are overlaid. The first histogram depicts the retinal interspike interval for the full retinal spike train (shown in blue). The second histogram depicts only the interspike intervals such that the second retinal spike drives an LGN spike (shown in orange).

The main takeaway from Fig 5B and 5C is that, in general, only those retinal spikes that come within quick succession of the previous retinal spike () are nearly guaranteed to generate an LGN output [5,6,8,11–13]. Matching experimental observations [5], Fig 5C shows a critical drop off of retinal to LGN spike transmission after the 15ms bin (). The histogram in Fig 5C, also shows that in the awake state there can be a high percent transmission of long ISIs () (even if they occur infrequently) [5,6]. Interestingly, this increased probability of successful transmission happens for retinal spikes near the beta (16–35Hz) frequency range, a set of frequencies implicated in LGN feedforward signaling to cortex [59,60].

The voltage trace in Fig 5 suggests a partial mechanism for each of the above results. The near faithful transmission of short ISIs and failure of medium length ISIs appears to be at least partially a result of EPSP summation, as noted in previous works studying anesthetized data [3,7,8,11]. In the case of , the voltage trace in Fig 5B hints that there has been sufficient decay of the membrane potential to steady state such that any input is enough to generate a spike. A natural follow-up is to ask which model components control these awake dynamics, and their subsequent observed relationships with retinal ISIs during retinal transmission. This relationship between retinal ISI, LGN spike probability, and the responsible ionic currents will be the focus of the next results subsection.

2.6. Filtering different retinal ISIs depends on the timescales provided by and

The first observed relationship between ionic currents and retinal filtering is that is necessary to produce the awake dynamics of states III and IV where filtering is actually possible. While both the potassium leak current () and the M current () (a non-inactivating slowly decaying potassium current) are controlled by neuromodulatory attention and arousal signals [32,33,44,50–53], the M current has a much stronger impact on the existence and transitions between the awake states. This is shown by contrasting the results of Figs 4 and 6, where Fig 6 plots the relationship between dynamic firing regions (Fig 6A) and the mean filtering properties (transfer frequency in Fig 6B and delay time in Fig 6C) as a function of and . Consistent with the findings for , the majority of parameter space that satisfies the filtering requirements overlaps with states III and IV (Figs 4D and 6D). However, unlike when is varied, Fig 6A shows that the existence of states III and IV requires a nonzero . This implies that for high acetylcholine (the main modulator of ), and thus low , state V becomes the dominant awake dynamic regime, meaning high arousal signaling ensures all inputs are successfully passed regardless of timing. Further differentiating from , increasing increases the width of states III and IV thus creating a larger range of external excitation values where retinal filtering is possible than was observed for increasing .

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Fig 6. M current is necessary for resting awake state dynamics and corresponding retinal filtering.

A) Depiction of the relationship between thalamocortical model state of arousal, cholinergic modulation of the M current conductance () and external excitation (). State labeling and curve color are identical to Fig 2. For those interested, S7 Fig depicts the complete bifurcation diagram with labeled mathematical bifurcation curves. B) Transfer ratio (colorbar value) between incoming retinal spikes and outgoing LGN neuron spikes plotted as a function of and . C) The delay time (colorbar value) between LGN spikes and the nearest driving incoming retinal spike plotted for the same and parameter space. D) Overlap between B and C where filtering occurs: transfer ratio <1 and delay time <5ms. The details for calculating transfer frequency and the delay time are provided in the Methods. Awake state transition curves (dashed black and purple lines) have been redrawn in Fig 6B-6D for further clarity.

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An examination of how the different ionic currents combine to determine the relationships between retinal ISI length and successful transmission reveals that and are the most important currents in the awake filtering process. Overlaying the LGN membrane potential with both the timing events of a retinal spike train, and each of the non-spiking, non-background ionic currents (Fig 7A), shows that and are the two most active of the four non-spiking currents. In the awake states the M current is always active and responds to all incoming retinal inputs with a rapid increase and a slow decay (Fig 7A). Meanwhile, is silent in the absence of inputs, activates if the TC cell crosses the spiking threshold, and inactivates rapidly without additional incoming input (Fig 7A). The other two currents are silent (, a hyperpolarization-activated cyclic nucleotide-gated current) or minimally active (, a low-threshold T-type calcium current) with or without retinal input (Fig 7A) due to the sufficiently depolarized membrane potential.

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Fig 7. The relationships between key ionic currents and the successful transmission of retinal spikes.

A) Membrane potential of the LGN neuron is shown in dark blue with corresponding values recorded on the left axis. Retinal spike times are denoted by red tick marks. The ionic currents , , , and are plotted in green, magenta, orange, and light blue respectively with values recorded on the right y-axis. B) Changes in the transmission probability of different retinal ISIs as a result of increasing (.12 to .165) (orange) or decreasing (.23 to .15) (pink) when compared with the control (black) parameter set’s response. In total, 10000 retinal spikes were simulated with ISIs grouped into 3ms bins. Plots here are for a TC cell in state IV, but the same trends are present for the less excited state III (see S8 Fig and Section 2.9). For comparison with experimental recordings of retinogeniculate transmission probabilities see Fig 9 in [5].

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Fig 7 reveals that the inactivation time constant of and the decay time of are key factors for determining the transmission probability in the awake state for short and medium length retinal ISIs respectively. At baseline, the TC cell shows a high passage of short (0–10ms) ISIs, a poor transmission of 10ms to 30ms ISIs (medium length), and good transmission probability for retinal ISIs beyond 30ms (long) (Fig 7B, black line). The activation of is partially responsible for the high passage of short ISIs while the inactivation time scale of dictates that this facilitatory window is between 0–10ms (Fig 7). The poor transmission of medium length ISIs is due to the slowly decaying perpetually active M current decreasing the membrane potential V fast enough such that the next incoming excitatory input is insufficient for V to cross the spiking threshold (Fig 7). After 30ms, the M current has decayed sufficiently following a thalamic spike such that long retinal ISIs can be transmitted. Increasing decreases the transmission probability for medium length ISIs by at least 20%, and up to 50%, while lowering decreases the transmission probability for short ISIs by (Fig 7B). One unexpected result from Fig 7B is that decreasing also increases the transmission probability for long ISIs. However, this is simply a byproduct of decreasing short ISI transmission: fewer clustered LGN outputs results in a smaller M-current increase thereby shortening the M-current imposed filtering window. Finally, note that the relationships between currents and filtering properties are consistent across dynamic regions III and IV (See S8 Fig and Section 2.9).

Combining the results of this subsection with previous subsections’ results we conclude that, in the awake state, the ionic currents of the thalamus are capable of filtering out, or relaying, inputs according to their timing. In agreement with experimental literature [5,7], the thalamocortical cell model in states III and IV filters out inputs received with ISIs between 15–30ms, corresponding to a gamma timescale. However, the model predicts that longer retinal ISIs (>30ms) may be successfully passed in these dynamic states. These two observations lead us to hypothesize that the thalamus may pass slower periodic frequencies and that faster gamma rhythms may be successfully passed only if they are embedded within a slower frequency oscillation. These investigations will be the subject of the following subsection.

2.7. Thalamus entrains to alpha and beta periodic frequencies but theta and gamma are best transmitted as nested oscillations

Upwards of 90% of the synapses made with LGN thalamocortical cells are modulatory, with inputs arriving from layer 6 in V1 cortex, the brainstem, the TRN, and local inhibitory interneurons [3,60]. Thus the LGN receives a variety of feedback inputs, many of which are oscillatory rhythmic signals normally generated within the noted brain areas including: theta (4–8Hz), alpha (8–14Hz), beta (14–30Hz), and gamma (30–100Hz) frequencies [61–64]. Signals occurring at these different frequencies, both as periodic inputs and nested inputs, have been implicated in numerous brain functions including: spatial memory (theta), attention (alpha and theta-gamma), working memory (gamma, theta-gamma) [65–68]. To generate predictions about how the awake thalamus may process non-sensory rhythmic signals occurring at these functional frequency bands, modulatory or otherwise, we stimulate the model with a variety of different rhythmic inputs. We stimulated the TC cell with periodic (5–70Hz) and nested (25–70Hz embedded within 4–12Hz) excitatory and inhibitory rhythmic signals. This subsection focuses primarily on signal processing during the alpha generating awake state IV (as this is the experimentally observed thalamic baseline awake state/rhythm), but will discuss rhythmic signal processing in states III and V at the end.

The model preferentially transmits alpha and beta excitatory periodic frequencies over faster gamma or slower delta/theta (1–7Hz) frequency inputs. Fig 8A shows an example of the thalamus successfully transmitting a high beta (27Hz) frequency input: in fact, inputs between 7Hz and 30Hz result in perfect entrainment (overlap with orange 1:1 line in Fig 8B), albeit with different output firing types (single spikes, doublets, or mixed mode) (S9 Fig). In contrast, for all frequencies between 30Hz and 70Hz, the ratio of input to output frequency falls beneath the orange 1:1 line, indicating that gamma frequency inputs are not faithfully transmitted. We find that the thalamic output is limited to frequencies below 30Hz in response to any periodic input at frequencies between 30Hz and 70Hz (Fig 8B). Finally, input frequencies in the theta range (4–7Hz) result in a partial 2:1 (output frequency is twice the input frequency) entrainment (overlap with green 2:1 line in Fig 8B), where additional non-stimulus driven spikes cause the output to fire at twice the input frequency, thereby failing to produce a theta rhythm.

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Fig 8. Awake thalamus successfully entrains to periodic excitatory alpha and beta rhythms but passes theta and gamma only as a nested rhythm.

A) Sample trace of the TC cell (in state IV) response to a periodic 27Hz spiking frequency input; the thalamus is perfectly entrained to the signal. Red tick marks denote timestamps of incoming excitatory spiking inputs. B) Summary plot showing the number of thalamic burst (or single spike) outputs per second in response to periodic rhythmic excitatory input. Line colors correspond to 2:1 partial entrainment (green) and 1:1 entrainment (orange). C) Sample trace of neuron’s response to a nested theta-gamma (8/66.5Hz) input showing near perfect entrainment to incoming gamma spikes. Entrainment depends on three features that are labeled here: the silent period between the incoming bursts of spikes (labeled as Delay), the combined frequency and number of the incoming fast inputs (labeled as Input Length) and the underlying dynamic state of the neuron (labeled as Underlying State). The impact of each of these features is analyzed/discussed within the text. Red tick marks denote timestamps of incoming excitatory spiking inputs. D) TC cell entrainment to the spiking gamma frequency inputs as a function of the incoming slow rhythm frequency. The dependent variable is calculated as the number of produced TC spikes that match the incoming gamma frequency divided by the number of incoming spikes at the gamma frequency. A value of 1 here indicates 1:1 entrainment to the gamma frequency inputs. Gamma frequency inputs are fixed at 66.5Hz, while the slow timescale (denoted by Delay in panel C) is varied to generate a slow rhythm frequency between 3Hz and 12Hz.

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While the model fails to pass excitatory periodic spiking gamma or theta faithfully, the thalamus can transmit these frequencies when they are nested. Fig 8C shows that, in response to an excitatory 66Hz spiking rhythm nested within an 8Hz oscillation, the thalamus entrains to the nested 66Hz signal. Excitatory gamma transmission is most successful when embedded within theta frequency oscillations and shows decreased effectiveness if embedded within alpha (>9Hz) or delta (<4Hz) rhythms (Fig 8D). Interestingly, within the theta frequency band, there are two distinct peaks for perfect 1:1 gamma entrainment found at 4-5.5Hz and 7.5-9Hz (Figs 8D and S10). This is noteworthy, as distinctions between high and low theta have been made previously in relation to working memory, behavioral tasks and spatial cognitive processes [67,69,70].

Since the thalamus also receives inhibitory feedback via the TRN and local interneurons, we repeat the above analyses for inhibitory signals, starting with periodic inputs. For periodic input frequencies in the range of 5–28Hz, the TC cell produces three distinct forms of entrainment: 2:1 partial entrainment (5–8Hz), 1:1 perfect entrainment (8–16Hz), and 1:2 (output frequency is half the input frequency) partial entrainment (18–28Hz) (Fig 9A and 9B). Although inhibitory and excitatory periodic inputs between 5–28Hz both produce a combination of full and partial entrainment, the thalamic output for inhibitory inputs is limited to between 8Hz and 16Hz. (Figs 8B and 9B). A noteworthy similarity between inhibitory and excitatory periodic inputs over their entrainment ranges is that, as input frequency is increased, the output firing mode changes from multi-spike to single spike (Fig 9A). In both cases, gamma frequencies (>30Hz) are not transmitted successfully in state IV. However, unlike the response to excitatory inputs, increasing the periodic inhibitory input frequency beyond 30Hz results in a sharply decreasing output frequency trending towards full output suppression (Fig 9B). Additionally, the phase relationship between periodic input and output is different for excitatory and inhibitory signals; for 1:1 entrained excitatory signals each input spike always sharply precedes an output spike, whereas the inhibitory signal occurs at varying phases in relation to the output spike, depending on input frequency (Figs 8B and 9A).

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Fig 9. Thalamus preferentially entrains (1:1) to alpha and beta periodic inhibitory rhythms while transforming nested theta-gamma rhythms into a faster frequency nested output.

A) Sample responses of a thalamic neuron to periodic inhibitory input: 1:1 entrainment with doublets (8Hz input, left), 1:1 entrainment with single spikes (15Hz input, middle), non-entrained output at a slower frequency (82Hz input, right). Red tick marks denote the timestamps of the inhibitory input spikes. B) Summary plot showing number of thalamic burst (or single spike) outputs per second in response to periodic rhythmic inhibitory input. Line colors correspond to 2:1 partial entrainment where output frequency is twice the input frequency (green), 1:1 pure entrainment (orange), and 1:2 partial entrainment where output frequency is half the input frequency (magenta) C) Number of thalamic input-driven bursts per second in response to a nested theta-beta frequency. Gamma frequency inputs are fixed at 35Hz, while slow theta timescale is varied. D) Responses of a thalamic neuron to 35Hz inputs nested in a 4Hz theta (left), nested in a 6Hz theta (middle), non-nested for comparison (right).

https://doi.org/10.1371/journal.pcbi.1014654.g009

If inhibitory spiking gamma inputs are coupled to a slow oscillation, the thalamus entrains to the slow frequency but not to the fast frequency. Provided a gamma (33Hz) is nested within a slow oscillation between 4Hz and 8Hz, the TC cell responds to the incoming slow rhythm with 2:1 partial entrainment (4-4.5Hz) or 1:1 entrainment (4.5-8Hz) (Fig 9C and 9D). Unlike with excitatory nested rhythms, when 1:1 perfect entrainment to the slow rhythm occurs, the spiking output is at a much faster frequency than the incoming gamma input (Fig 9D). Furthermore, as was observed with inhibitory periodic gamma frequency inputs, the nested spiking inputs have a different timing relationship to the outputs than was observed with nested excitatory rhythms (Fig 9D); instead of each input spike preceding an output spike, the inhibitory input spikes are grouped and precede the full set of grouped output spikes. Thus, while inhibitory theta nested gamma inputs can be used to entrain to the slow rhythm, the relationship between the incoming gamma spikes and the gamma frequency of the output spikes is vastly different from the response to excitatory nested inputs. Note that, for an identical non-nested gamma (33Hz) frequency input, the TC cell output is a 13Hz alpha rhythm, indicating that nesting an incoming gamma within a slower rhythm is crucial for theta generation (Fig 9D).

While the above simulations for nested rhythms were all run for fixed gamma inputs (4 spikes at 66Hz for excitatory or 4 spikes at 35Hz for inhibitory), faithful transmission of the theta rhythm, with a singular clear nested gamma frequency output, demands a fast enough spiking nested input. Notably, for excitatory nested inputs, the percent entrainment to the incoming gamma frequency, and percent transmission of total inputs regardless of output frequency, is maximized for faster (66Hz and 100Hz) versus slower (40Hz and 50Hz) gamma frequencies with fewer spikes per slow cycle (S11 and S12 Figs). Inhibitory nested rhythms also require a threshold for the gamma frequency, with entrainment to the theta rhythm occurring only for frequencies faster than 30Hz (S12 Fig). Interested readers are directed to the supplement for a more detailed analysis of the relationship between the nested gamma input properties and the thalamic output.

Perturbations of the underlying dynamic state suggest that state IV is optimal, when contrasted with states III and V, for entraining to slower (alpha and beta) periodic excitatory rhythmic signals. More specifically, shifting to a faster high-threshold bursting underlying frequency by increasing excitation (moving from state III to state IV through increases of or decreases of ) results in a broader range of excitatory periodic input frequencies that the model can entrain to. Across the entire parameter space devoted to both regions III and IV, the lower bound for entrainment is roughly in the theta band (6–8Hz) until the IV-V border (Fig 10A). However, moving from state III into state IV, the upper bound of entrainment drastically increases from near 15Hz (region III sample point) to near, or above, 25Hz (region IV sample point) (Fig 10B). Note that once the model reaches state V, both the minimum and maximum entrainment frequencies drastically change and are constrained to be in the gamma frequency (30–70Hz), no longer entraining to alpha or beta (Fig 10A and 10B).

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Fig 10. Possible entrainment range widens as the underlying dynamics of the model are moved from state III to state IV.

A) Lower frequency limit that the model is capable of entraining to for excitatory periodic inputs. B) Upper frequency limit that the model is capable of entraining to for excitatory periodic inputs. C) Same as A, but for inhibitory periodic signals. D) Same as B, but for inhibitory periodic signals. All entrainment frequencies (given in Hz) are presented as color intensity (depicted on colorbar) for varying values of and potassium leak conductance (). All awake state regions (III, IV and V) and delineating borders are redrawn for clarity. A representative point in each region is highlighted for ease of description and comparison of the entrainment properties within and between regions. Model response to periodic inputs outside of the entrainment limits in states III and IV can be found in S13 Fig. Note the differences in color bar values when comparing subplots A and C with subplots B and D.

https://doi.org/10.1371/journal.pcbi.1014654.g010

For periodic inhibitory signals across regions III and IV, we find that increasing external excitation () increases both the upper and lower bound for possible entrainment frequency. The lower bound is increased minimally from HZ (region III sample point) to (region IV sample point) (Fig 10C). However, the upper bound is limited to Hz in region III but can reach HZ in region IV (Fig 10D). This implies that an increase in baseline high-threshold bursting frequency (moving from state III to IV) broadens the range of possible entrainment frequencies, critically allowing for the passage of inhibitory low beta signals. Once again, note that upon transition to state V, there is a large shift in entrainment properties: the model no longer entrains to inhibitory theta or alpha frequencies, but can reproduce an incoming inhibitory gamma (Fig 10C and 10D).

Regarding nested inputs, although all three awake states produce a theta rhythm in response to excitatory and inhibitory theta-gamma inputs (S14 and S15 Figs), there are major differences between the spiking outputs. For instance, in state V, both excitatory and inhibitory nested inputs results in the output of two distinct spiking gamma frequencies, while only one gamma frequency output is generated in states III and IV (S14 and S15 Figs). Furthermore, while inhibitory nested inputs appear to produce a similar output in states III and IV, only state IV successfully transmits all of the incoming excitatory gamma spikes (S14 Fig).

These results, coupled with the results from previous subsections, imply that arousal state IV, the state producing the baseline awake alpha rhythm, is an ideal sensory filtering state and is also optimal for rhythmic transmission.

2.8. Implications for neuropsychiatric diseases

The L-type calcium current gene CACNA1 is one of many risk genes for a variety of psychiatric disorders including schizophrenia [71,72]. It has been proposed that mutations of this gene could be partially responsible for the modulation of EEG alpha to theta/delta frequencies in the awake state [73]. Here we explore the effects that a known Single Nucleotide Polymorphism (SNP) mutation of the CACNA1 gene, which increases the L-type calcium channel conductance in induced (reprogrammed stem cells) human neurons [74], might have on the thalamus and thalamic processing.

The model suggests that increasing alters the underlying baseline firing properties and eliminates the robust entrainment features of the state IV thalamus. The first observation from Fig 11A is that the increase in L-type calcium conductance changes the baseline high-threshold alpha frequency bursting of state IV into a bursting output occurring at a theta frequency. While the theta bursting output has a faster ISI and more spikes per burst in comparison with the high-threshold alpha, it also does not fully resemble low-threshold bursting, implying an intermediate bursting type. This alteration in firing behavior also affects the response of the thalamus to periodic excitatory and inhibitory inputs. While the unaffected state IV TC cell produced 1:1 entrainment to frequencies from 7-30Hz (excitatory) and 8–16Hz (inhibitory) (Figs 8 and 9), here the model fails to entrain to any input above 12Hz (Fig 11B). Furthermore, in the altered TC neuron, the output frequencies (y-axis value) are restricted to values between 4Hz and 12Hz, regardless of input type (excitatory or inhibitory) and frequency (Fig 11B). This result suggests that this genetic mutation in the L-type calcium channel gene CACNA1 could lead to a rigidity in thalamic processing that may help explain schizophrenia related deficits in cognitive flexibility and task switching [75,76].

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Fig 11. Increases in L-type calcium conductance () change the baseline firing type and entrainment properties of the state IV thalamus.

A) Membrane potential plots showing the underlying state dynamics and response to periodic excitatory inputs (at 24Hz) for an unaltered thalamocortical neuron (top) and a thalamocortical neuron with increased (bottom). External spiking input is turned on after 2000ms, with the timing of the inputs denoted by the red tick marks. B) Summary plot showing number of thalamic input-driven burst (or single spike) outputs per second in response to incoming periodic rhythmic excitatory (blue circles) or inhibitory (green diamonds) signals. These responses are for a neuron initialized in state IV but with increased .

https://doi.org/10.1371/journal.pcbi.1014654.g011

2.9. Model is most sensitive to changes in and

This section will address model robustness and sensitivity to changes in parameters for states III and IV (high-threshold bursting states), which we have shown emerge as the dynamic working states. Readers interested explicitly in sleep dynamics corresponding to states I and II in our model, and their dependence on parameters, are directed to previous modeling studies [31,34,77]. The central result of this analysis is that the alpha high-threshold bursting of the model and its associated functional properties are most sensitive to changes in and , currents implicated in neuromodulatory signaling pathways and disease/disorder developments (see previous subsections).

We first investigate how the novel high-threshold bursting properties (burst frequency, ISI, minimum voltage) depend on the conductance values of the four non-spiking, non-background currents ( and ). To do so, we select four model parameters sets from within regions III and IV of the vs parameter space (to show these results hold for different combinations of excitation and neuromodulation levels) and then independently vary the listed conductances (see Methods).

Fig 12 shows that high-threshold bursting is robust to changes in and , but sensitive to changes in and . Variations in or by 10% resulted in less than a 5% change for all three of the high-threshold burst properties. However, varying by 10% resulted in a near equivalent percent change (with opposite sign) in the high-threshold bursting frequency. Notably, we find the model was most sensitive to changes in ; a 5% change in consistently resulted in a drastic shift in firing properties, or eliminated high-threshold burst firing altogether (Fig 12 and S1 Table). These results are striking as is modulated during attention and arousal while is implicated in the development of neuropsychiatric diseases (see previous subsection). The extreme sensitivity of model properties to changes in and suggests that the thalamus tightly regulates these currents to allow for different properties depending on functional context. For example, high sensitivity of the model to and may permit alterations of the filtering window or entrainment ranges established by these two currents to flexibly enable the passage of signals which are normally not transmitted.

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Fig 12. High-threshold bursting exhibits high sensitivity to changes in and but not or .

The average (across four parameter sets) percent change (%) from the recorded value for a control parameter set of each of the three key high-threshold bursting properties (frequency:blue, ISI:red, and minimum voltage:yellow) is plotted as a function of manipulated ionic conductance. Conductances were varied in both directions (labeled here as Inc for increased or Dec for decreased). The full non-averaged data set containing the values for each of the four chosen parameter sets is presented in S1 Table.

https://doi.org/10.1371/journal.pcbi.1014654.g012

Sections 2.4-2.6 characterize mean retinal transmission properties and how the filtering of different retinal ISIs is influenced by and ; here we show that these features are robust to changes in and , and that the conclusions of sections 2.4-2.6 hold under different levels of baseline model excitation (different parameter sets from the vs parameter space are chosen). S16A Fig shows that the transmission probability curve for different length ISIs is largely unaltered by changing either or . More specifically, under these parameter variations, the model still predicts a high passage of short and long ISIs with a strong filtering out of the medium length retinal spike times. Additionally, although the total transmission probability increases with increased excitation (moving from state III to state IV), the timescales of retinal ISIs preferentially passed through the thalamus are similar throughout regions III and IV of vs parameter space with the greatest percentage of filtered out ISIs occurring in the 20–40ms range (S16B Fig).

These sensitivity analyses, in tandem with results of the previous subsections, show that all three emphasized model properties explored here (dynamic firing state, retinal filtering, rhythmic entrainment) show some robustness to variations in parameters within states III and IV. We see that regions III and IV both operate, and respond to parameter changes, similarly, with the caveat that a more depolarized neuron (further into state IV) passes (or entrains to) more signals (both rhythmic and non-rhythmic). And finally, across all analyses, we find the model properties are sensitive primarily to changes in the M current and L-type calcium current, further highlighting the importance of these channels for awake thalamocortical signal processing and their possible impacts on disease development.

3.1. Summary of results

We constructed a biophysical conductance-based LGN model, with model components constrained using genomic databases [32,33]. The additions of the M-current () and the L-type calcium current () to the canonical currents from previous thalamic models [34,35,77], were critical for reproducing the full range of experimentally observed LGN firing dynamics. In particular, in the absence of noise, the interaction of and produces a firing state of alpha frequency high-threshold bursting. Our model suggests that this dynamic state does not require a special thalamocortical cell subtype as suggested by previous works [18,35,36]. In total, the model finds five different firing modes, each of which can be associated with a state of arousal (Fig 1) [18,37,38]. Two of these LGN states (I and II) are associated with low-threshold bursting sleep dynamics, the two new states (III and IV) correspond to the theta and alpha rhythms seen from freely moving animals during natural wakefulness [78], and the final state (V) describes the most excited state of focused attention. Importantly, the model transitions between these different states dependent on neuromodulatory arousal signals and/or external excitation (Fig 2).

The model reveals that incoming retinal signals interact with the five thalamic states in different ways; retinal spike train inputs are largely ignored during sleep-like states (states I and II), filtered during the theta and alpha generating awake dynamics (states III and IV), and faithfully transmitted only during the most excited state V (Fig 3). Notably, the area of parameter space where the model exhibits retinal filtering precisely overlaps with the newly modeled dynamic regimes III and IV specific to our model (Figs 4 and 6). Our model’s version of filtering compared with key retinal transfer studies in awake and anesthetized animals shows agreement across the mean and granular properties of retinal transmission (Fig 5) [5,7,9,15]. Our analysis shows that within the novel states III and IV, the model reproduces the experimentally observed heterogeneity in the mean percentages of transmitted retinal spikes. A deeper examination shows that the timescales governing and interact with the timing of retinal inputs to specifically control which retinal ISIs are faithfully transmitted in the awake states (Fig 7).

Finally, the model examines how the thalamus might process and interact with non-sensory signals coming from separate brain areas such as the cortex, basal ganglia, and thalamic reticular nucleus. This was done by considering a wide variety of different incoming rhythmic oscillations that have been suggested to play a significant role in inter-area brain communication. Input types investigated include both periodic, and nested, excitatory and inhibitory rhythmic input (Figs 8 and 9). Our analysis revealed that in the filtering states thalamic 1:1 entrainment is possible for periodic excitatory and inhibitory inputs at alpha and beta frequencies but that the passage of periodic gamma and theta frequency inputs is limited. Successful transmission of both gamma and theta frequency inputs was achieved when a spiking gamma signal was embedded within a slower theta cycle. Investigating rhythmic signal transmission during the different awake states suggests that state IV is optimal for rhythmic signal processing (Fig 10). We demonstrate that perturbations from this state in manners consistent with different neurological disorders leads to significant decreases in thalamic processing power. For example, we show how an experimentally observed alteration of the L-type calcium current associated with schizophrenia disrupts the awake alpha rhythm thereby limiting the rhythmic entrainment range of the thalamus (Fig 11).

3.2. Neuromodulatory inputs are necessary to understand the complete dynamic range of the thalamus

Neuromodulatory signals from the brainstem and cortex have been shown experimentally to be critical in transitioning the thalamus between sleep and waking states [24,46,48,79,80]. The depolarizing effect of both glutamatergic and cholinergic signaling is modeled here by decreasing potassium conductances, or , known downstream targets of activated mGlurRs and mAChRs [35,42,48]. Interestingly, it has been proposed that these two receptor types have different somatodendritic distributions that could partially account for some of the differences between glutamatergic and cholinergic signaling in the LGN [37]. In our model we find that differences in neuromodulatory signaling could be based on the downstream ion channels that are impacted. Although and both help control the level of depolarization in our model, unlike , also controls the existence or absence of the alpha high-threshold bursting state and its associated filtering properties. Our results suggest that under different levels of neuromodulatory tone, an identical range of external excitation can span different subsets of the five dynamic states and the transitions among them (Figs 4 and 6). This observation helps provide two possible explanations for why high-threshold bursting was not observed experimentally in some thalamocortical LGN cells [28]: either in such cells is extremely low (due to poor expression or high neuromodulatory tone) such that states III and IV don’t exist for any value of , or the strength of the neuromodulatory tone is low enough that states III and IV are not reached by the tested levels of applied current (Figs 4 and 6).

While this work specifically emphasizes cholinergic and glutamatergic signaling, the thalamus also receives a variety of other neuromodulatory inputs including norepinephrine, histamine, serotonin and adenosine signals [18,35,81]. Some neuromodulators, such as histamine, reduce , and thus are expected to exert a similar effect on the LGN as acetylcholine does [18,82]. Neuromodulators can also target other ionic conductances, such as [41]; it has been shown that manipulations of conductance have the capacity to shift the rhythmic firing dynamics during the low-threshold bursting associated with sleep [77]. In total, state dependence on ionic conductance levels suggests that the full characterization of neuronal dynamics seen in vitro, and more importantly in vivo, requires neuromodulator application during experimental studies.

3.3. M and L-type calcium currents help structure awake retinal transmission

In the natural wakefulness states, the M current works to prevent medium length (ms) retinal ISIs from generating LGN spikes while simultaneously allowing some of the longest length retinal intervals (ms) to be transmitted. This implies the LGN in the resting awake state could prevent low gamma frequency inputs (30–80Hz) from being passed while allowing the passage of beta rhythms. These results help suggest a reason for why, in monkey LGN, it was observed that V1-LGN communication happens in the alpha/beta frequency bands [60,61]. Within the model, we see a high percent passage of these low gamma frequency retinal inputs only when the model has reached the highest arousal state (Fig 3). This indicates that additional excitation (such as increased cholinergic tone or cortical feedback signals) are required to transmit gamma band activity [18], a key EEG signal of increased attention and arousal [40,59].

The model also suggests that may be a key determining factor in the experimentally observed paired spike facilitation (increased probability of successful transmission when the incoming retinal input closely follows a previous input) [7]. Specifically, L-type calcium currents work to permit clustered LGN spikes in response to retinal inputs with the shortest ISIs (0–10ms) with almost 100% faithful transmission (Fig 7). This relationship of retinal ISIs with the L-type calcium current could offer further explanation about a noted difference between awake versus anesthetized animals: the duration of the facilitation effect of retinal spike pairs is shown to be shorter for awake animals compared with anesthetized ones [5]. One possible explanation for this is that the hyperpolarizing effect of anesthetics [83] deactivates the L-type calcium channels, preventing them from helping to pass clustered retinal spikes. Also some of the anesthetics used in retinogeniculate transfer studies, such as propofol, have the capability to selectively block L-type calcium currents [84,85]. In the model, lowering the L-type calcium conductance () decreases the percent of transmitted short ISIs (Fig 7). This, combined with an unexpected increase in the passage of long ISIs, results in an increased average transmitted ISI (Fig 7). The resulting impact of changing the transmitted retinal spike distribution due to decreased is a decrease in paired spike facilitation and a less dependent relationship between short retinal ISIs and successful transmission.

3.4. Translating an LGN model to other thalamic nuclei

Although our model is constrained based on available LGN data, the intuition gained here about the distinct dynamic states and their filtering properties may also be relevant for understanding how signals are processed in other thalamic nuclei. This includes investigating how auditory and somatosensory inputs are handled by the medial geniculate and ventral posterior thalamus respectively, as well as deciphering how higher order nuclei such as the pulvinar and mediodorsal thalamus process signals related to more complex cognitive functions like attention [86–88]. We expect similar results when extending (and modifying) our model to non-LGN thalamic nuclei as brain atlas databases [32,33] confirm the presence of the currents used in our model in other thalamic nuclei across different species. This critically includes the M-current and L-type calcium current which, in the more depolarized states of our model, are responsible for creating dynamic states III and IV, selectively passing tightly clustered inputs and periodic inputs at a beta (or lower) frequency while simultaneously preventing the throughput of tonic low gamma frequency inputs.

However, there are some noted experimental differences between the LGN and other thalamic nuclei to consider for future modeling endeavors. It has been observed that the LGN has a higher density of cholinergic fibers compared to other primary sensory nuclei [89], indicating the relationship between , and dynamic state may be altered when modeling the auditory and somatosensory thalamus. Furthermore, for translating to higher order nuclei it has been noted that, in comparison to lower order thalamus, these nuclei produce low-threshold bursts in the awake state more frequently, and in some cases hyperpolarize, instead of depolarize, in response to cholinergic signaling [88,90]. The increased presence of low-threshold bursting in higher order thalamus is particularly interesting as our model indicates that this type of bursting prevents the transmission of non-rhythmic signals. This implies that the low-threshold bursts in higher order nuclei may be acting as a synchronizing or attention signal instead of facilitating active signal processing [88]. This connection between low-threshold bursting and attention was recently examined directly in the pulvinar nucleus using electrical microstimulation, where it was revealed that pulvinar bursting enhanced target detection and broadly increased control of cortical processing and behaviors [91]. This observed importance of low-threshold bursting in higher order cortical control may suggest that the baseline working state for higher order nuclei is state III (from which you can reach the low-threshold bursting state II more easily) and not state IV, as is proposed here for lower order nuclei. In total, the experimentally observed differences between higher order and lower order nuclei indicate that model alterations may be required to properly capture awake processing in higher order thalamus. This may entail adding additional ionic currents, changing the existing model ionic current conductances and/or including different (non-cholinergic/glutamatergic) neuromodulatory signals thereby allowing awake higher order nuclei to still enter state IV and enable the optimal signal processing and filtering predicted to occur there.

3.5. Re-framing the awake alpha rhythm

The awake alpha rhythm was the first recorded human brain rhythm and has been observed repeatedly in numerous species [36,92–94]. The origins of the awake alpha rhythm are still hotly debated. Some studies have attributed the generation of alpha to the cortex [23,95], others relate awake alpha power to distinct nuclei of the thalamus (such as the LGN) [36], and others consider alpha to arise as an interaction of the two [21,96]. It remains an open question if and how these mechanisms interact [23]. Our model predicts that in the absence of external inputs the thalamus generates a clear alpha that may be influential in driving a cortical V1 alpha. We also predict that the thalamus will entrain to corticothalamic feedback at alpha frequencies depending on the shape (number of spikes per alpha rhythm peak) of the alpha input (S11 Fig). In tandem, these predictions suggest a context and signal dependent coexistence of the thalamic and cortical alpha rhythms that could be tested in a future cortico-thalamo-cortical model.

Functionally, alpha rhythms have been hypothesized to: act as an idling state, increase in power with memory load or memory maintenance, exhibit elevated power during task distractor stimuli presentation, have a causal role in the allocation of spatial attention, and be inversely correlated with neuronal spiking and gamma band activity [92,97–101]. Here we propose a new functional understanding (consistent with the above interpretations) of the thalamic alpha rhythm: this rhythm is a marker that the thalamus is in a working state prepared to filter both rhythmic and non-rhythmic incoming input; the alpha rhythm itself is not explicitly processing, or suppressing, information through the thalamus. Instead, the model state IV (where the baseline rhythm is alpha) is where and are interacting to produce filtering and entrainment capabilities; it happens that in the absence of external signals this combination is producing an alpha.

Our model has the following important properties: it produces alpha in the absence of inputs, it exhibits decreased alpha in response to sensory and non-alpha-frequency rhythmic signals, and it entrains to periodic alpha frequency inputs. These properties are in agreement with experimental observations and can be used to provide mechanistic explanations. Examples are: why the alpha rhythm is largest during periods of eyes closed relaxed wakefulness and is sharply decreased upon eyes opening [102] (an absence of sensory inputs into our model increases alpha); why alpha amplitude decreases when attending to a stimulus [22,103] (an increase of external inputs to our model decreases alpha); and why alpha is increased during memory maintenance [104] (a top down cortical-produced alpha seen during memory maintenance entrains thalamic output in our model, thereby increasing alpha power).

The model decrease in baseline alpha in response to external signals may have an analogue in the decreases of baseline beta (11–35Hz) power found in macaque [105]. Again, it may not be the frequency that is causal, but the interactions between the currents and (possibly with altered time constants or conductance values) that are causal. We therefore predict that manipulations of the conductances and kinetic properties of and that disrupt the baseline alpha (or beta) rhythms will also disrupt many of these functional properties currently attributed to these rhythms as well.

3.6. Extending beyond retinogeniculate transmission: rhythms and disease

Our model TC cell is capable of generating a variety of experimentally observed rhythms at different frequencies including: low-threshold bursting delta (1–4Hz) and theta (4–8Hz) rhythms during sleep, single spike and high-threshold bursting theta/alpha (4–14 Hz) rhythms associated with natural wakefulness, and fast tonic spiking at beta/gamma (20–80Hz) [18,27,31,37]. Alterations in these rhythms have been associated with diseased states. For example, a number of neurological disorders categorized as thalamocortical dysrhythmias (TCDs), including schizophrenia, Parkinson’s disease, absence epilepsy, tinnitus, and depression have been shown to exhibit decreases in thalamocortical alpha power with a corresponding increase of theta or delta power [21,106]. While TCD associated disorders span multiple sensory modalities (motor, auditory and visual), thereby involving different thalamic nuclei, the underlying disrupted electrophysiological processes may be similar [107,108]. In some cases, the increase in the slower rhythms has been associated with increased bursting, changes in HCN channel expression, and/or chronic deinactivation of calcium channels [25,77,108].

Our modeling results predict that, by slowing the underlying thalamic frequency from alpha to theta (or delta), neurological disorders will heavily impact the ability of the thalamus to process external stimuli. First, if the thalamus is perturbed towards state III from the alpha bursting state IV, the model exhibits a notable decrease in transmission of non-rhythmic excitatory inputs (Fig 3). Furthermore, in the awake theta generating state III, the model thalamus also displays a decreased ability (compared to state IV) to entrain to both excitatory and inhibitory rhythmic frequencies outside of the theta frequency band (Figs S13 and 10). This further exacerbates the increase in theta power associated with thalamocortical dysrhythmias [21,107], as theta becomes the primary thalamic output, regardless of input signal, in this underlying state. We observe a similar response to rhythmic inputs where the thalamic outputs are restricted to the theta band (Fig 11) upon perturbation of the L-type calcium current in a manner consistent with the schizophrenia literature [71,74]. Limited thalamic flexibility to entrain to rhythmic feedback signals may help explain the observed attention deficits in schizophrenia patients [75,76,109] and why L-type calcium channel blockers such as isradipine have displayed success in improving cognitive function in schizophrenia [110]. Finally, if a thalamocortical dysrhythmia perturbs the baseline firing mode all the way to state II (low-threshold bursting at theta), the model predicts a complete inability to process non-rhythmic stimuli. Deciphering which baseline model theta-producing state (state III, L-type perturbed state, or state II) corresponds to the diseased phenotype in question will help uncover mechanisms behind the dysfunction and propose suggestions for restoring the thalamus to its optimal functional state.

The TCD slowing of baseline alpha to theta also affects how the thalamus processes nested oscillations. Coupled with the observed increase in theta, many of these dysrhythmias are also associated with increases in beta and gamma [25,111,112]. These observations are in agreement with our model results, as both the resting theta output in state III, and the theta output generated in response to incoming theta nested gamma rhythms, take the form of high frequency bursts (happening at beta and gamma frequencies) nested within theta oscillations (Figs 2 and 9). Interestingly, unlike in state IV (underlying alpha), in state III (underlying theta), when the thalamus is entraining to an excitatory theta nested gamma, the output gamma spikes do not match frequency, or entrain, to the input gamma signals. This inability to entrain to the incoming gamma in state III (the expected state seen during dysrhythmia phenotypes) could further negatively impact the ability of the thalamus to interpret external signals.

Outside of their relationships with TCDs, experimental work within and outside of the thalamus suggests that gamma nested slow oscillations may play an important role in thalamic communication and coordination with other brain areas. Theta-gamma oscillations are most prominently studied in the hippocampus for their positive relationship between cross-frequency-coupling and memory recall [113,114]. Recent work has also shown evidence that theta-gamma coupling in the anterior thalamic nucleus is important for memory encoding [115], while pulvinar alpha can be coupled to cortical gamma influencing network synchrony and attentional selection [116]. Our modeling results suggest that gamma-nested slow oscillations (theta/alpha) may be possible within multiple thalamic nuclei, and that entrainment is likely dependent on the initial dynamic behavioral state. We find that entrainment success depends on the nested frequencies of the input signal as well as the phase alignment between the underlying dynamics and incoming signals, a feature controlled in state IV by the interactions of the M current and L-type calcium current (S10 Fig). Interestingly, these two currents are both expressed in brain areas such as the hippocampus and neocortex [32,33], areas where theta-gamma coupling is prominent and has been attributed to properties of the M-current [117]. In addition to the theta-gamma coupling, in our model the interactions of the M and L-type calcium currents produce excitatory and inhibitory periodic entrainment in the alpha and beta bands; future studies could examine whether the interactions of these expressed currents in the hippocampus and neocortex might display similar periodic entrainment results.

3.7. Comparison with previous modeling studies

A variety of different conductance-based thalamic models (cellular and network) have been used previously to study rhythmic oscillations under different arousal states. Many of these models [118–122] were constructed to study low-threshold bursting properties of the thalamus associated with sleep rhythms (Slow-wave and delta), Up and Down States, resonance and implications for epilepsy. However, these models were limited in their ability to reproduce the full range of awake LGN states seen by the Crunelli group [26,27]. The vast majority of previous thalamocortical models, including the pioneering models by Destexhe and McCormick [31,77,118–124], produce only a single tonic spiking firing mode associated with the awake thalamus. Meanwhile, models by both Vijayan and Kopell [35], and Li et. al [18]., created two subtypes of TC cells, one of which could produce the alpha high-threshold bursting state, but failed to produce the slow tonic firing regime. Furthermore, each of these later high-threshold bursting models hypothesized the existence of a high-threshold T-type calcium current in a specialized thalamocortical cell that still lacks experimental validation. By comparison, our model employs currents that have been validated against genomic databases [32,33], produces all experimentally found firing types, and correctly transitions between all five arousal states in a manner consistent with changes in external excitation and neuromodulatory signaling [21,46,47,125]. Additionally, contrasted with the Li et. al. model [18], our TC cell, starting in the awake alpha state, exhibits a broader range of entrainment to excitatory frequencies, most notably exhibiting successful passage of the beta frequency band. LGN entrainment to beta is a particularly important feature of the model as beta frequency band feedforward and feedback interactions are essential to LGN-V1 communication [59,61]. Finally, in this alpha state, our model reveals that TC cells are capable of relaying certain periodic inhibitory rhythms, as well as passing high frequency theta-nested gamma excitatory and inhibitory rhythms, types of rhythmic phenomena not explored in the previous thalamic models [18,31,35,77].

Researchers have also used mathematical modeling to examine retinogeniculate transfer, although these studies have been mostly restricted to using firing rate, integrate-and-fire, or statistical models [8,11–13]. Collectively, these models confirmed that retinal ISI length was the biggest determining factor in successful retina-LGN transmission in anesthetized animals, and that the process could be approximated with an exponential filter [13]. However, based on the employed model type and data used, these models were not intended to examine retinal spike transmission in awake animals and how it changes with arousal level. This limitation precludes addressing which components of the neuron physiology are responsible for determining the differences in transmitted retinal ISIs when comparing between awake and anesthetized data, namely: the higher average efficacy in wakefulness, the decreased duration of facilitating effects, and the increased transmission of longer retinal ISIs in wakefulness [5]. Here, our detailed conductance-based model addresses these issues by tuning parameter values to control arousal state while simultaneously examining the impact of other ionic conductances on retinal transmission. Specifically, our model framework helps reveal that modulation of the potassium leak and M-current conductances ( and ) controls arousal level and thus the average transfer efficacy, while the timescales of (L-type calcium current conductance) and are responsible for the decreased spike facilitation and the increased transmission of longer ISIs in awake animals. One possible extension for this model to consider is the idea of directly modeling extra-retinal inputs instead of considering them to be the action of the parameter . These extensions could help elucidate the role of cortical (or TRN) feedback in retinogeniculate transfer, a process that is still under investigation, but has been discussed in previous works [13] with modeling attempts using some of the simplified frameworks listed earlier [17,126].

3.8. Caveats and limitations

Our model used a minimal subset of expressed thalamic ion channels that can produce key experimentally observed in-vitro dynamics. We note that this chosen subset by no means exhausts what has been found experimentally [33], or used in previous computational models [18,31,118–122]. For example, previous sleep-focused thalamic literature highlights the inward rectifier potassium current () and the Calcium-activated Non-selective cation current () as having roles in determining sleep characteristics (Up and Down state waveforms, plateau potentials, afterhyperpolarization [118–122]) not fully encoded by our model. However, removing details (currents) that we believe to be extraneous for our questions concerning awake thalamic processing helped create better clarity for understanding what we propose are the underlying mechanisms generating the awake dynamics and filtering/entrainment properties while still preserving necessary basic features associated with sleep (calcium-driven bursting, bursting at <1 and delta frequencies).

Although the kinetic equations for the channels in our model have been fit to experimental data, in some cases, these data and their corresponding fits, were derived from non-LGN thalamic nuclei [31,127] or from different brain regions entirely such as the hippocampus, the olfactory bulb, or the cortex [18,31,128–131]. Additionally, many of these recordings have been performed across a wide range of species. These limitations highlight the need for experimentalists to continue performing in-vitro clamp recording experiments for fitting accurate detailed conductance-based models that produce mechanistic insights and experimentally testable predictions in the brain regions and test species of interest.

In this study we used a cellular model specifically to enable comparisons with the experimental retinogeniculate transfer studies (and their associated data). By using a single cell model, when examining the ability of the model to filter retinal signals, we can only consider postsynaptic effects and how a given incoming retinal spike interacts with the existing active ionic currents. Although this ignores the possible influence of presynaptic effects like facilitation and depression [9], it has been observed that retinogeniculate synaptic amplitude remains constant across different ISI lengths, implying that the presynaptic effects are likely minimal [8].

Previous network models of the thalamus have postulated that alpha-bursting TC cells can phase the outputs of non-bursting cells [30,35]. Because the alpha bursting cells in these network models provide a driving alpha signal unaffected by feedback signals, we predict that the same results would emerge if our TC cells were used to produce the alpha in these network models instead. However, in general, these alpha producing TC cells are connected with gap junctions and are receiving feedback inhibition (from the TRN and local interneurons) and feedback excitation (from the cortex). These interactions introduce additional time scales that filter or amplify a given signal in a manner distinct from the proposed processing of rhythmic signals we investigated in section 2.7 [3,13,29]. To fully explore the impacts of these network level effects would require a network structure.

5.1. Conductance-based model experimental background

To examine properties of retinogeniculate transmission in different states of arousal (i.e., deep and light sleep, natural wakefulness, attentive awake), we need a model that distinguishes between different LGN states. Due to the historical focus of experimental literature [1–3,77,132], previous conductance-based computational models emphasized replicating sleep rhythms and their dynamics. These experimental studies prompted the development of the model by Destexhe et al. [77], that is still widely used as a modeling baseline today. Within this model, a low-threshold T-type Calcium current and a hyperpolarization-activated cation current combine to produce the low-threshold calcium bursting dynamics associated with sleep rhythms. The remaining currents in the model are the spiking sodium and potassium channels (, ) and two different leak currents and . While this combination of currents suffices for sleep dynamics, it is not sufficient to capture the complexity of awake firing.

Originally, the awake state LGN (and other thalamic nuclei) were believed to exhibit primarily tonic spiking interspersed with occasional low-threshold bursts [1–4]. While attempting to understand the natural wakefulness associated alpha rhythm in the thalamus [21,29,36,37], experimentalists found a novel awake state firing mode termed high-threshold bursting. In the LGN, it was shown that a subset (10–30 percent) of excitatory TC cells generated this novel form of bursting with an interburst interval in the alpha (8–14Hz) frequency range [21,26,27,36]. Crunelli et al. reproduced this bursting phenomenon in slice experiments after the application of mGluR1 or mAChR agonists [21,29,30,36], that are known to affect the conductances of the background potassium channels [42,48,49,125]. Due to distinct properties of high-threshold bursting (discussed later in the Methods), Crunelli et. al. hypothesized that a different (still unknown) mechanism is responsible for this bursting type versus the generators of low-threshold burst firing [27].

Using these experimental results, some computational models have incorporated this novel firing behavior into their investigations about awake thalamic dynamics [18,35]. However, the lack of clarity surrounding the high-threshold bursting mechanism resulted in key model limitations [18,35]. Specifically, these models required a phenomenologically constructed calcium current, and were able to replicate only a subset of the full spectrum of awake dynamics [18,35]. These issues could lead to incorrect dynamic behavior when analyzing and interpreting visual signal processing. Based on these drawbacks, and our intention to study retinogeniculate transmission at different levels of arousal, there was a need for a novel thalamocortical cell model that has experimentally validated currents, and the capability to produce the full breadth of thalamic dynamics.

5.2. Model formulation

This subsection presents our single cell lateral geniculate thalamocortical conductance-based mathematical model:

(1)

Equation 1 is formulated as a Hodgkin-Huxley style single compartment model [133]. The model dictates that changes in cellular membrane potential V are the result of fluxes through membrane bound ionic channels () and external forcing/noise to the cell () scaled by the inverse of the capacitance . To replicate known experimental observations, and generate experimentally testable and functionally relevant predictions, we constrain the selection of currents for using expression data sourced from the Allen Brain Institute and other public databases [32,33]. More specifically, we look up the gene associated with each ion channel chosen to be included in the model and verify positive expression within the LGN and the thalamus more broadly. It is important to highlight that we are not modeling/including ALL channels expressed within the thalamus, but are instead searching for a minimal subset that reproduces the dynamics of interest.

Validating our selections against this expression data, we arrive at the following minimal model for :

(2)

where the specific current kinetics are explicitly detailed in the Supplement (S1 Text). In Equation 2 there are four groups of current types; the first 3 are widely used across previous thalamic modeling studies [18,31,34,35,77,124,134]. First we have the spiking sodium () and potassium () currents necessary to produce action potentials [133]. The model also includes 3 background currents: leak , potassium leak , and persistent sodium to set baseline neuronal excitability levels [18,133]. Then, to capture sleep dynamics, the third set of necessary currents, (a hyperpolarization-activated cyclic nucleotide-gated current) and (T-type calcium channel), are included [1,2,77]. The final two channel types are the non-inactivating Muscarinic potassium current and a high-threshold L-type calcium channel . Based on previous cortical modeling studies [34,43], we hypothesized that these currents could be used to generate high-threshold bursting in the thalamus. Note that in comparison with existing thalamic models, and are relatively novel additions, and are not ubiquitous across LGN TC cell models [35,49,77,124]. A summary diagram of our TC cell model and the primary purpose for each of the chosen ionic currents is provided in the supplement (S17 Fig). Note, this selection provides a minimal subset sufficient for capturing our dynamics of interest, but additional currents highlighted in other studies may be useful for future analyses [18,31,124].

5.3. Model fitting

To address retinal transmission across different arousal levels, the model first needs to properly reproduce the full observed range of experimental firing types. This is accomplished by first running a parameter search algorithm to find possible candidate models that produce the two most distinct firing behaviors: classic low-threshold bursting seen during sleep, and the novel high-threshold bursting phenomenon associated with the awake thalamic alpha rhythm (highlighted by the work of the Crunelli Lab) [28,36,37]. Critically, our algorithm requires that the candidate model generates both types of bursting from a single parameter set with variations only to external excitation levels. Good candidate models were determined based on their fit to an objective function that quantitatively described the low- and high- threshold bursting dynamics. Following the completion of the initial parameter search, each candidate model was then subjected to a series of stress tests including: varying to check for the existence of the other dynamic regimes and manipulations of key ionic conductances to replicate pharmacological blocking experiments (Figs 1 and S2). The results presented within this manuscript are for one candidate model produced via the parameter search algorithm that successfully reproduced the full set of experimental results of the Crunelli lab [28,36,37]. The remainder of this subsection describes the specific details associated with this parameter search algorithm including the parameter set chosen and the constructed objective function.

The parameter search was conducted by using MATLAB’s built in genetic algorithm [135]. This algorithm searches through a bounded, pre-defined parameter space by minimizing a given objective function that we define to have a minimum when the model produces low-threshold bursting for a small value of and high-threshold bursting for a larger value of . The parameter space used here consists of a subset of parameters from of Equation 1. Note that the intrinsic ionic currents of presented in Equation 2 have the general structure:

(3)

with maximal conductance , gating variables m and h, and reversal potential . By enforcing that the gating kinetics and reversal potentials to be fixed based on their experimentally fit values from literature [31,127,133,136–138], our parameter space was restricted to the 9 ionic conductance values and . The upper and lower bounds for the conductances set during the parameter search were provided by previous modeling studies of the thalamus and the cortex (in the case of the M-current where previous thalamic modeling studies are lacking) [18,31,34].

The objective function that the algorithm minimized to search through parameter space was based on work by Marder et. al. (and others) [139–141]. More specifically, the weighted objective function is constructed to quantitatively describe both high- and low-threshold bursting dynamics. Based on experiments by the Crunelli group, three defining bursting characteristics were focused on for the construction of this objective function [27,28,37]. The first distinguishing feature is bursting frequency. While there are a variety of different rhythms that have been recorded during sleep (low-threshold bursting) and waking (high-threshold bursting) states [27,28,38], two well known frequency bands (1–4Hz delta for sleep and 8–14Hz alpha for wakefulness [18,38]) are used here. The second characteristic built into the objective function captures how the inter-spike intervals within a burst are different depending on burst type. High-threshold bursts are restricted to have fixed ISI lengths in the range of 10–20ms [27,30,36], while low-threshold burst ISIs are more variable and also much shorter [27]. The last piece of our objective function concerns a minimum observed membrane potential. For the awake high-threshold bursting, we restrict the membrane potential to stay above -60mV to agree with the experimental results [36]. This is a critical distinguishing feature as it dictates which currents are involved in the bursting mechanism and separates high-threshold bursting from low-threshold bursting because of the clear hyperpolarization events.

Combining these three descriptors of bursting qualities results in a weighted objective function consisting of 5 components: frequency of low-threshold bursts , frequency of high-threshold bursts , interspike interval within a low-threshold burst , interspike interval within a high-threshold burst , and the minimum observed voltage during high-threshold bursting . This combination produces the error function given below:

(4)

Each term of Equation 4 is constructed such that the objective function is minimized for a window of values for the quantitative feature: 1Hz-3Hz , 8Hz-12Hz , 5ms-9ms , 14ms-18ms and -57mV- -53mV to enable model flexibility. Specific functional forms can be found in the attached supplemental code (See subsection 5.7).

5.4. Retinal spike transmission

Examining the LGN cell model response to visual stimuli required a mathematical formulation of retinal input. A simple spiking retinal ganglion cell (RGC) was synaptically connected to provide fast excitation to the LGN TC cell. An RGC current () was appended into Equation 1 where is the strength of the synapse and its reversal potential. The synapse is controlled by the gating variable , whose dynamics are adopted from [69] to give:

(5)

with decay constant , rise constant and retinal ganglion cell membrane potential . For this study, the RGC’s membrane potential is set to a maximum on each spike, and then decays according to the following equation:

(6)

Retinal spike timing in this model was determined by choosing a mean firing rate (60 spikes per second unless noted otherwise) and then sampling spike times from a corresponding Gamma distribution with the given mean [5,10–12].

Analyzing the input-output relationship between the spiking retinal cell and the LGN output spikes was done by recording a few key experimental metrics. The primary recorded metric is retinogeniculate efficacy (percent transmission) defined as the ratio of output LGN spikes to incoming retinal spikes [5,7–9,13]. To verify that LGN spikes are being driven by the incoming retinal spikes the corresponding delay time between LGN spike time and the preceding retinal spike time is recorded. In accordance with experimental literature a RGC is considered to be driving LGN spiking provided the the delay time between spikes is between 0–5ms [7,9,13].

5.5. Rhythmic signal transmission

The retinal spiking Equations 5 and 6 are repurposed to investigate how the thalamic neuron responds to rhythmic signaling. Rhythmic excitatory feedback inputs are modeled using the fast excitatory synapse employed for the RGC spiking inputs. For inhibitory inputs, the parameters , and were adjusted accordingly to reflect a fast inhibitory synapse [69]. Nested rhythmic inputs for excitatory and inhibitory signals are determined by dictating the inter- and intra- burst ISIs as well as the desired number of spikes per input burst.

5.6. Sensitivity and robustness analysis

Four representative models were chosen from within regions III and IV of the - parameter space of Fig 2 to examine feature specific fluctuations in response to parameter variations. For all four chosen models, each of four key parameters ( and ) were varied individually by of their original value ( was used in instances where destroyed high-threshold bursting) to determine model feature sensitivity. The feature values computed were the frequency of bursting, the ISIs of the bursts, and the minimum observed voltage value: the three values emphasized within previous experimental studies for defining high-threshold bursting [27]. For each varied parameter, the percent change from the baseline value exhibited by the control parameter set was computed for all three electrophysiological features.

We also repeated the filtering and entrainment simulations for multiple representative models chosen from throughout regions III and IV of parameter space to show the relationship with underlying state. For retinal filtering, the percent transmission of simulated retinal spike trains for twelve representative models (from the - parameter space) was recorded as a function of retinal ISI. A similar procedure was used for the entrainment simulations; each different model (according to position in - parameter space) received the full range of periodic inhibitory and excitatory spike trains. The upper and lower limits of the frequency where pure 1:1 entrainment occurred were recorded. In this case, entrainment was defined to be where the difference between the input frequency and the primary model output frequency (recorded as spikes or burst events per second) was less than one.

5.7. Simulation replication and data availability

Except for the two-parameter bifurcation curves, all other simulations were run in MATLAB [135]. The ODE system was solved using MATLAB’s built in numerical solvers. The two parameter bifurcation curves were made using the numerical bifurcation and continuation package in XPP-AUTO [142]. Note that in all bifurcation diagrams (and associated colormap plots) the potassium conductances and have been scaled (by .06 and .3 respectively) for numerical simulation and visualization purposes.

All code can be accessed at: https://github.com/keesmcgahan/Thalamocortical-Highthreshold-Bursting-Cell-Model

All of the ODEs, current equations, kinetic activation and inactivation rates and additional parameter values can be found in the supplement (S1 Text) as well as within the accompanying code files.