A spiking corticothalamic model recreates key features of cellular physiology

The dynamical elements of the model were inspired by recent empirical observations, and consist of three classes of neurons (Fig 1A and 1B) – L5_PT cells (blue), fast spiking interneurons (basket cells, gold), and diffuse projecting matrix thalamocortical cells (purple) – each of which are modelled using biophysically plausible spiking neurons [34–37] that were coupled through conductance-based synapses. By building the model from these circuit elements, we ensured that the emergent dynamics of the population recapitulate known signatures of cell-type-specific firing patterns, thus retaining the capacity to translate insights between computational modellers and cellular physiologists.

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Fig 1. Thalamocortical model of perceptual awareness. A) Idealised anatomy of the model thalamocortical loop connecting higher-order matrix thalamus and L5_PT neurons. B) Single neuron dynamics of example neurons in the thalamocortical network for each class of cell when driven with 600 Hz of independent background drive to the somatic compartment of every neuron and 50 Hz to apical compartment of L5_PT cells. C) Bifurcation diagram of the L5_PT apical compartment. I_B1 denotes a saddle node bifurcation generating a stable plateau potential which coexists with the resting state of the apical compartment until the model passes through a second saddle node bifurcation at I_B2 at which point the resting state vanishes and the stable plateau potential becomes globally attracting. D) Somatic firing rate of the novel dual compartment L5_PT model as a function of basal (i.e. somatic compartment) drive and apical drive (measured in terms of the distance to I_B1). In line with empirical data [33], apical drive increases the gain of the somatic compartment. E) Model thalamocortical ring architecture. L5_PT cells and basket cells were placed on a cortical ring at evenly space intervals.

https://doi.org/10.1371/journal.pcbi.1012951.g001

To model the non-linear bursting behaviour of L5_PT cells (which has been linked to perceptual awareness [33,38,39]), we created a novel dual-compartment model with active apical dendrites that captures the essential features of the cells’ physiology. Empirical recordings have shown that L5_PT cells switch from regular spiking to bursting when they receive near-simultaneous input to both their apical (top) and basal (bottom) dendrites ([33,38]; Fig 1B). As such, our model includes a somatic compartment, described by an Izhikevich adaptive quadratic integrate and fire neuron [36,40,41] and an apical compartment, described by a non-linear model of the Ca²⁺ plateau potential [37,42]. To recapitulate known physiology, we coupled the compartments such that sodium spikes in the somatic compartment back propagate to the apical compartment; in turn if a Ca²⁺ plateau potential is triggered in the apical compartment the somatic compartment’s behaviour (probabilistically) switches from regular spiking to bursting (implemented by switching the reset conditions of the somatic compartment). The amount of current entering the apical compartment controls this switching process. With sufficiently high current the apical compartment passes through a saddle node bifurcation (I_B1; Fig 1C) and a stable Ca²⁺ plateau potential coexists with the resting state of the compartment. Further increases in current cause the cell’s resting state to disappear (by passing the cell through a second saddle node bifurcation at I_B2; Fig 1C), making the plateau potential globally attracting (Fig A in S1 Appendix).

Based on the finding that communication between the soma and apical dendrites of L5_PT cells depends upon depolarising input from the matrix thalamus to the “apical coupling zone” of L5_PT cells [43], we made the probability of successful propagation between compartments proportional to the amplitude of thalamic conductances. In line with empirical findings and previous modelling [33,44], Ca²⁺ plateau potentials in the apical compartment controlled the gain of the somatic compartment’s firing rate curve by increasing the amount of time the somatic compartment spent in a bursting rather than a regular spiking parameter regime (Fig 1D).

In line with previous spiking neural network models of early sensory cortex [25,26,45,46], we embedded the cortical neurons in a one-dimensional ring architecture (90 pairs of L5_PT excitatory and fast-spiking inhibitory interneurons; Fig 1E). Each point on the ring represents an orientation preference, with one full rotation around the ring corresponding to a 180^° visual rotation – this provides each neuron with a 2^° difference in orientation preference, relative to its neighbours. The cortical ring was coupled to a thalamic ring with a 9:1 ratio (to approximately reflect the cortico-thalamic ratio in mammals), which then projected back up to the apical dendrites of the same 9 cortical neurons, representing the diffuse projections of higher-order thalamus onto the apical dendrites of L5_PT neurons in layer 1 [47,48]. Cortical coupling was modelled with a spatial decay, with long range inhibitory coupling and comparatively local excitatory coupling (i.e., centre-surround ‘Mexican-hat’ connectivity).

When driven solely by baseline input, the model emitted irregular spikes interspersed with sparse spatially localised bursts mediated by depolarising input from the thalamus which allowed L5_PT somatic spikes to back-propagate initiating Ca²⁺ plateau potentials in the apical dendrites which in turn initiated transient burst spiking in the somatic compartment (Fig 1C). Cortical spiking activity was highly irregular (mean inter-spike interval coefficient of variation = 2.4) characteristic of a waking state [49]. For more details on the model architecture and analysis of the dynamics see materials and methods.

The thalamocortical model reproduces empirical signatures of threshold detection

We first set out to reproduce the results of the whisker-based tactile detection paradigm employed by Takahashi and colleagues [12,13], who trained mice to report a mechanical deflection of a whisker over a range of deflection intensities (Fig 2A) while recording L5_PT activity in barrel cortex from the apical dendrites via fast scanning two-photon Ca²⁺ imaging, and somatic activity via juxtacellular electrodes. They found that bursting activity in the soma of L5_PT cells, generated by Ca²⁺ plateau potentials in the apical dendrites, distinguished hits and false alarms from misses and correct rejections. Importantly, they were able to establish causality through a series of perturbation experiments (Fig 2B). Optogenetic excitation of the apical dendrites reduced the animal’s threshold for awareness increasing both hits and false-alarms (Fig 2C). In turn, pharmocological inhibition of the apical dendites and POm (a matrix-rich higher-order thalamic nucleus with closed loop connections to barrel cortex [47]) increased the animals perceptual threshold (Fig 2D and 2E).

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Fig 2. Apical compartment distance to bifurcation and thalamic-gating explains shifts in perceptual threshold. A) Whisker deflection paradigm of Takahashi and colleagues [12]. B) Representation of causal perturbations to L5_PT thalamocortical circuit including optogenetic excitation of the apical dendrites (light blue), pharmacological inhibition of the apical dendrites (pink), and pharmacological inhibition of POm (orange). C-D) Psychometric function of animals performing a whisker deflection task for the control condition (grey), optogenetic excitation of apical dendrites (blue), and pharmacological inhibition of apical dendrites (pink). Modified from Takahashi and colleagues [12,13]. E) Response probability as a function of whisker deflection intensity for control (grey) and pharmacological inhibition of POm (orange). Modified from Takahashi and colleagues [12]. F-H) Model response probability as a function of stimulus intensity across simulated causal perturbations. Colours same as above. I) Apical compartment bifurcation diagram showing the average distance to B₁ in post stimulus period averaged across stimulus intensities. J) Average distance to B₁ across stimulus intensities and simulated causal perturbations in the post stimulus period. K) Proportion of population in bursting regime across stimulus intensities and simulated causal perturbations in the post stimulus period. L) Average inter-compartment coupling probability across stimulus intensities and simulated causal perturbations in the post stimulus period. M) Average spike count across stimulus intensities and simulated causal perturbations in the post stimulus period.

https://doi.org/10.1371/journal.pcbi.1012951.g002

To model the perceptual discrimination process underlying threshold detection –discriminating the presence of a weak stimulus against a noisy background – all neurons received a constant background drive consisting of independent Poisson spike trains while an arbitrary cortical neuron was pulsed by a current of constant width and variable amplitude that we weighted by a spatial Gaussian to mimic the selectivity of neurons in early sensory cortex. We operationalised perceptual awareness in the threshold detection simulations in terms of what a downstream ideal observer could readout from the population by computing whether trial-by-trial spike counts in the 1000 ms post stimulus window exceeded an optimal criterion (i.e. the criterion that best minimised misses and false alarms across stimulus intensities). We counted the model as having made a response whenever the spike count exceeded the optimal criterion. Psychometric functions were then fit to the model’ responses (for details see materials and methods). Qualitatively identical results were obtained using neurometric functions which summed over all criterion values (see Fig B in S1 Appendix [50]).

The model responses to simulated whisker deflections recapitulated the empirically observed sigmoidal relationship between the intensity of the simulation and the model’s response probability (Fig 2F-H). In addition, perturbations to the model designed to replicate optogenetic excitation and pharmacological inhibition (Fig 2A) qualitatively reproduced the empirically observed shifts in response probability across perturbation types (Fig 2C-H). For each type of perturbation we ran the simulation over a range of perturbation magnitudes to ensure the reliability of the effect see Fig B in S1 Appendix. For brevity, we only show the results for 300 pA perturbations in the main text. In addition, we note that the saturating detection probability values in the model are due to an absense of behavioural stochasticity resulting from extranious factors such as decision noise which are inherent to empirical data.

A key benefit of biophysical modelling is the capacity to mechanistically probe the model and determine how the empirical observations may have emerged from the underlying circuit dynamics. To this end, we used tools from dynamical systems theory to interrogate the cell-type-specific dynamics underlying the behaviour of the model including the distance to bifurcation (I_B1) in the apical compartment, the parameter regime of the somatic compartment (i.e. regular spiking or bursting), and the inter-compartment coupling probability. Excitation of the apical dendrites (blue) reduced the average distance to bifurcation (here defined as the distance to I_B1) in the apical compartment across the network in the 1000 ms period post stimulus onset (Fig 2I and 2J). This increased the proportion of time each somatic compartment spent in the bursting regime (Fig 2K), which in turn increased the average spike count of L5_PT cells across stimulus intensities resulting in reduction of the model’s perceptual threshold (Fig 2F and 2M). Conversely, inhibition of the apical dendrites (pink) and the thalamus (orange) both resulted in an increase in the perceptual threshold (Fig 2G and 2H).

Importantly, however, the mechanisms underlying the increase in the perceptual threshold differed across apical dendrite and thalamic inhibitory perturbations. Inhibition of the apical dendrites increased the average distance to bifurcation at B₁ in the apical compartment across the network (Fig 2I and 2J). In contrast, inhibition of the thalamus resulted in a comparatively minor reduction in the distance to bifurcation in the apical dendrites (Fig 2I and 2J) but reduced the thalamus mediated inter-compartment coupling (Fig 2L) thereby reducing the probability that a back propagating action potential could reach the apical compartment, and the probability with which a plateau potential could switch the regime of the somatic compartment from regular spiking to bursting. Together this resulted in a similar reduction in the proportion of cells in the bursting regime for both thalamic and apical dendrite inhibition, and likewise, a similar reduction in the average stimulus evoked spike count explaining the comparable increase in perceptual thresholds (Fig 2M).

Thalamocortical spiking model generalises to visual rivalry

We next sought to generalise our thalamocortical model of perceptual awareness to visual rivalry formalising and interrogating the hypothesis that the role played by pulvinar – L5_PT loops in visual cortex is analogous to the role played by POm – L5_PT loops in barrel cortex. To simulate visual rivalry we drove the model with input representing orthogonal gratings presented to each eye, typical of standard binocular rivalry experiments [21,51], targeting the soma (i.e., basal dendrites) of L5_PT cells on opposite sides of the ring with orthogonal orientation preferences (Fig 3A).

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Fig 3. A thalamocortical cascade underlies perceptual switches. A) Thalamocortical ring architecture driven by input representing orthogonal gratings. B) Raster plots of L5_PT soma (blue) and matrix thalamus population (purple) during rivalry. C) Example single neuron spiking activity around a perceptual switch – dotted lines denote the crossing in the population averaged time series. D) Population averaged neuronal variables centred on a perceptual switch. E) Histogram of dominance durations, black line shows the fit of a Gamma distribution with parameters estimated via MLE (α = 4.85, θ = 0.56). F) Simulation confirming Levelt’s second proposition. Dashed line shows the dominance duration of the population receiving the decreasing external drive, solid line shows dominance duration of population receiving a fixed drive. G) Simulation of Levelt’s fourth proposition. Notably, for low stimulus strengths our model, along with a number of meanfield models of visual rivalry [28], predicts a deviation from Levelt’s fourth proposition.

https://doi.org/10.1371/journal.pcbi.1012951.g003

Due to the fact that inhibitory connectivity is broader than excitatory connectivity, delivering external drive to opposite sides of the ring shifts the model into a winner-take-all regime with burst-dependent persistent states on either side of the ring competing to inhibit one another. Importantly, through the accumulation of slow a hyperpolarising adaptation current in the somatic compartment of each L5_PT cell (representing slow Ca²⁺ mediated K⁺ currents [52,53]), burst-dependent persistent states are only transiently stable leading to stochastic switches between persistent states characteristic of binocular rivalry (Fig 3B and 3C). We operationalised perceptual dominance (i.e., awareness of a percept at the exclusion of the other) in terms of the difference in average firing rate between the persistent ‘bumps’ of L5_PT cell activity on opposite sides of the ring. In line with common practice (e.g [54]), a population was counted as dominant when it had a firing rate 5 Hz greater than its competitor and lasted for longer than 250 ms (results were robust across a large range of threshold values). The initial competition for perceptual dominance was determined by which population of L5_PT cells first established a recurrent loop with matrix-thalamus. This interaction allowed the recurrently connected population of cells to enter a bursting regime, at which point the population had sufficient activity to maintain a persistent state and inhibit the competing population into silence. In previous work we have argued that this emergent property of corticothalamic loops is a good candidate for a cellular-level correlate of perceptual awareness [10,15].

Crucially, our model provides a cellular-level explanation of spontaneous rivalry-induced perceptual switches in terms of these same mechanisms. As the reliable initiation of Ca²⁺ plateau potentials in the model depends upon back-propagating action potentials, perceptual switches are always preceded by regular spikes in the suppressed population. Preceding each switch, the accumulation of adaptation reduces the firing rate of the dominant population to a sufficient level that the suppressed population can escape from inhibition and emit a brief series of regular spikes that transition into bursts as thalamus is recruited and Ca²⁺ plateau potentials are initiated (Fig 3C), eventually gaining sufficient excitatory activity to inhibit the previously dominant population into silence via recurrent interactions with the basket cell population. At the population level (Fig 3D), this cascade of neuronal events is characterised by an initial ramping in the somatic firing rate of the suppressed population, followed by the inter-compartment coupling probability, the proportion of the population bursting, and finally by the average distance to bifurcation in the apical compartment. Once dominant, approximately half the population is in a bursting regime at each point in time (0.5424 mean  ±  0.058 standard deviation), and the average distance to bifurcation in the apical compartment fluctuates around zero (-11.0185  ±  48.067 pA), with approximately a third (0.3669  ±  0.097) of the population located above the critical boundary (I_B1) at each point in time. In addition, the average inter-compartment coupling probability is (0.8845  ±  0.102) allowing reliable communication between compartments. Once silenced, adaptation in the previously dominant population decays back to baseline levels and accumulates in the previously suppressed, now dominant, population. In this way, the competitive organisation of the cortico-cortical and corticothalamic loop provides a natural “flip-flop” switch that stochastically alternates between dominant percepts.

As cortical pyramidal cells are known to display considerable differences in spiking behaviour across species [55] we swept the parameters of our model L5_PT cell responsible for the bursting behaviour, and the inter-compartment coupling probability, to ensure that visual rivalry (quantified by average dominance duration) is stable across a wide range of parameters. In favour of the robustness of the model, as long as there was reliable inter-compartment coupling, which we assume is the case in the waking state, dominance durations in an empirically plausible range (i.e. with a period on the order of seconds) were present across a wide range of parameter values (Fig C in S1 Appendix).

In close agreement with psychophysical data, switches between dominance and suppression had a right-skewed distribution of dominance durations ([51,56,57]; Fig 3E). Across stimulus drive conditions (1300 – 1500 Hz), comparison of negative log likelihoods showed that the distribution of dominance durations was best fit by a Gamma distribution (), compared to lognormal () or normal distributions (). Because of the large size of the network and the discontinuity in the somatic compartment we could not use analytic methods to interrogate the structure of the dynamical system underlying the stochastic oscillations. Instead we used a heuristic line of argument (see [58], Ch.8) combined with simulations in the absence of noise [59] to confirm that the dynamical regime underlying these stochastic oscillations likely consists of noisy excursions around a stable closed orbit (i.e. a stable limit cycle; Fig D in S1 Appendix).

Thalamocortical spiking model conforms to Levelt’s propositions

To further test the psychophysical validity of our neurobiologically detailed model, we simulated the experimental conditions described by Levelt’s modified propositions [56] – a set of four statements that compactly summarise the relationship between stimulus strength (e.g., luminance contrast, colour contrast, stimulus motion) and the average dominance duration of each stimulus percept. Here, we focus on the modified second and fourth propositions, as they constitute the “core laws” of rivalry and incorporate recent psychophysical findings (propositions one and three are consequences of proposition two [56]).

Levelt’s modified second proposition states that increasing the difference in stimulus strength between the two eyes will principally increase the average dominance duration of the percept associated with the stronger stimulus [56,60]. To simulate Levelt’s second proposition, we decreased the spike rate entering one side of the ring from 1400 to 1250 Hz in steps of 50 Hz across simulations. In line with predictions (Fig 3F), the average dominance duration of the percept corresponding to the stronger stimulus showed a steep increase from ~3 s with matched input, to ~ 6.5 s with maximally different input while the average dominance duration on the side of the weakened stimulus decreased comparatively gradually to ~2 s with maximally different inputs.

According to Levelt’s modified fourth proposition, increasing the strength of the stimulus delivered to both eyes will increase the average perceptual reversal rate (i.e., decrease dominance durations [56]), a finding that has been replicated across a wide array of experimental settings [61–64]. To simulate Levelt’s fourth proposition, we ran a series of simulations in which we increased the spike rate of the external drive in steps from 1300 to 1500 Hz in steps of 50 Hz across simulations. Again in line with predictions (Fig 3G), the perceptual alternation rate increased with input strength, starting at ~7.75 alternations per minute at the second weakest stimulus strength (1350 Hz) and increasing to ~15 alternations per minute for the strongest stimulus (1500 Hz). Interestingly, along with a number of meanfield models of rivalry [28], our model predicts a deviation from Levelt’s fourth proposition for very low stimulus values with an uptick in alternation rate occurring at the lowest external dive value (1300 Hz). Encouragingly, there is some initial evidence that deviations from Levelt’s fourth law may be present in human psychophysical data [56].

To help ensure that the simulation results were not biased by finite size effects or other simplifying assumptions such as the all-to-all connectivity of the cortical ring, or the 50/50 excitatory/inhibitory neuron ratio, we show in Fig E in S1 Appendix that a scaled-up version model consisting of 2000 cortical neurons with sparse connectivity, and an 80/20 excitatory/inhibitory neuron ratio (i.e. consistent with Dale’s law), also produces a Gamma distribution of dominance durations, and is consistent with Levelt’s second and fourth propositions.

We thus confirmed that our neurobiologically detailed model of the matrix thalamus - L5_PT loop is capable of reproducing Levelt’s propositions, which together with the right-skewed distribution of dominance durations, show the consistency of our model with the psychophysical “laws” known to govern visual rivalry.

Generating testable predictions through in silico electrophysiology

Binocular rivalry is thought to depend in part on the substantial degree of binocular overlap in humans (~120^o), however the lateral position of the eyes in mice leaves only ~40 ^o of binocular overlap [65]. For this reason, there are no current mouse models of binocular rivalry, however there are monocular variants of visual rivalry, namely plaid perception, that can be studied the mouse model [7,66]. Crucially, plaid perception, like binocular rivalry, conforms to Levelt’s laws [56,67] and also has a right skewed distribution of dominance durations that is well fit by a Gamma distribution [66]. We hypothesise, therefore, that the principles underlying the simulation of binocular rivalry in our model will also describe other forms of visual rivalry (such as plaid perception), offering a plausible means to test cellular level predictions derived through simulation. To this end, we next ran a series of perturbation experiments, with the aim of interrogating the novel burst-dependent mechanism of perceptual dominance by mimicking the optogenetic and pharmacological experiments carried out in threshold-detection studies [12,13], in the context visual rivalry. As perceptual dominance depends on the formation and maintenance of a burst-dependent persistent state, we hypothesised that artificially exciting the apical dendrites would result in an increase in the average dominance duration of the excited population, and artificially inhibiting the apical dendrites and thalamus would result in a decrease in the dominance durations for the inhibited populations.

Due to the fact that the distance to bifurcation of the dominant population fluctuates around zero, we predicted that exciting the apical dendrites would increase the proportion of the population above the bifurcation at B₁, thereby reducing the probability with which a fluctuation in somatic drive would lead to a sizable enough drop in the proportion of the population below B₁ to release the competing population from inhibition. This should, therefore, result in an increase in the frequency of long dominance duration events, thereby increasing the mean and the spread of the distribution. Equivalently, we predicted that inhibiting the apical dendrites would reduce the proportion of the population above B_1, making it more likely that transient fluctuations in somatic drive would allow the competing population to escape from inhibition, reducing the occurrence of long dominance duration events, thereby reducing both the mean and the spread of the distribution of dominance durations. Finally, based on the results of the threshold detection simulations we predicted that thalamic inhibition would have an analogous effect on dominance durations to apical dendrite inhibition but would be mediated by a reduction in the coupling probability. To test these hypotheses, we conducted two in silico experiments analogous to the conditions described by Levelt’s modified propositions but instead of manipulating the external drive entering the somatic compartment of L5_PT cells we manipulated the amplitude of simulated causal perturbations to the L5_PT apical compartment and thalamus (Figs 4A and 5A).

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Fig 4. In silico electrophysiology reveals degenerate mechanisms of perceptual dominance (asymmetric perturbations). A) Causal perturbations to one half of the thalamocortical circuit underlying visual rivalry consisting of optogenetic excitation of the apical compartment (blue), pharmacological inhibition of the apical compartment (pink), and pharmacological inhibition of the thalamus (orange). B-D) Average dominance duration of perturbed (solid), and unperturbed (dashed) populations. Error bars show SEM. E) Average distance to bifurcation point at B₁ shown on bifurcation diagram for perturbed (solid) and unperturbed (dashed) populations during periods of perceptual dominance with 400 pA perturbation strength. F) Average distance to bifurcation point at B₁ for perturbed (solid) and unperturbed (dashed) populations during periods of perceptual dominance. G) Proportion of population above bifurcation point at B₁ during periods of perceptual dominance. H) Proportion of population in bursting regime during periods of perceptual dominance.

https://doi.org/10.1371/journal.pcbi.1012951.g004

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Fig 5. In silico electrophysiology reveals degenerate mechanisms of perceptual dominance (symmetric perturbations). A) Causal perturbations to full thalamocortical circuit underlying visual rivalry. Colours same as above. B-D) Perceptual alternations per minute of simulation time across perturbation types. Error bars show SEM. E) Average distance to bifurcation point at B₁ shown on bifurcation diagram for perturbed (blue, orange, pink) and unperturbed (grey) simulations during periods of perceptual dominance at 400 pA. F) Average distance to bifurcation point at B₁ for full network perturbations during periods of perceptual dominance. G) Average proportion of population above bifurcation point at B₁ for full network perturbations during periods of perceptual dominance. H) Average proportion of population in bursting regime for full network perturbations during periods of perceptual dominance.

https://doi.org/10.1371/journal.pcbi.1012951.g005

In the first set of experiments, we simulated optogenetic excitation and pharmacological inhibition of one of the two competing populations by adding a constant current ( 200, 400 pA) to all of the target variables (i.e. apical compartment or thalamic neurons) on one side of the ring (Fig 4A). In line with predictions, we found that the average dominance duration of the excited population (Fig 4B) increased, the distance to bifurcation decreased (Fig 4E and 4F), the proportion of the population above the critical point at B₁ increased (Fig 4G), and the proportion of the population in the bursting regime increased (Fig 4H). The dominance durations and neuronal dynamics of the unexcited population remained relatively unchanged.

Similarly, inhibition of both the apical dendrites and thalamus reduced the average dominance duration of the inhibited population while the uninhibited population was again relatively unchanged (Fig 4C and 4D). As in the threshold detection simulations, inhibition of the apical dendrites led to a large increase in the distance to B₁ compared to thalamic inhibition (Fig 4E and 4F) which primarily affected the inter-compartment coupling probability (Fig FA in S1 Appendix). Both apical dendrite and thalamic inhibition led to almost identical reductions in the proportion of the population above the critical point at B₁ (Fig 4G), and the proportion of the population in the bursting regime (Fig 4H). The uninhibited population again remained relatively constant across all of the neuronal measures (the small drop in dominance durations of the uninhibited population for the 400 pA inhibitory perturbations is due to the adaptation variable having less time to recover). As predicted the spread of the distribution of dominance durations increased with the amplitude of excitatory perturbation and decreased under inhibition (Fig FB in S1 Appendix).

In the second set of experiments, we simulated optogenetic excitation and pharmacological inhibition of both competing neuronal populations simultaneously by adding a constant current ( ±  200, 400 pA) to all of the target variables on the ring (Fig 5A). Again, in line with predictions, the speed of rivalry (i.e., the number of perceptual alternations per minute of simulation time) decreased as a function of apical dendrite excitation (Fig 5B). Excitation also decreased the average distance to B₁ (Fig 5E and 5F), increased the proportion of the population above the critical point at B₁ (Fig 5G), and increased the proportion of the population in the bursting regime (Fig 5H).

In contrast, inhibition of the apical dendrites and thalamus both increased the speed of rivalry (Fig 5C and 5D). Inhibition of the apical dendrites led to a large increase in the distance to the critical point at B₁ compared to thalamic inhibition (Fig 5E and 5F), but both apical dendrite inhibition and thalamic inhibition reduced the proportion of the population above the critical point at B₁ (Fig 5G), and the proportion of the population in a bursting regime (Fig 5H) through reductions in the thalamus mediated inter-compartment coupling probability (Fig FC in S1 Appendix). As with the asymmetric perturbation simulations, inhibition exerted a much larger effect on the speed of rivalry than excitation. Finally, again in line with our predictions, the spread of the distribution of dominance durations increased under excitatory perturbation of the apical dendrites and decreased under inhibition of the apical dendrites and thalamus (Fig FD in S1 Appendix).

Together, these in silico electrophysiological experiments provide important testable (and explainable) hypotheses for future experiments that although not testable in any existing data sets are well within the purview of modern systems neuroscience providing an opportunity to conduct precise theory-driven tests of the model.

Thalamocortical spiking neural network

The neuronal backbone of the model consists of a (novel) dual compartment model of L5_PT neurons, fast spiking interneurons (basket cells), and thalamic cells. The dynamics of basket cells, thalamic cells, and the somatic compartment of L5_PT cells (Fig 1A-C) were described by Izhikevich quadratic adaptive integrate and fire neurons, a hybrid dynamical system that is capable of reproducing a wide variety spiking behaviour while still being highly efficient to integrate numerically [34–36]. The Izhikevich neuron consists of the following two-dimensional system of ODEs:

(1)

(2)

with reset conditions: if then . The equations are in dimensional form giving the membrane potential (including the resting potential spike threshold , and spike peak , and reset c), input , time t, and capacitance C biophysically interpretable units (mV, pA, mS, and pF respectively). The remaining four parameters k, a, b, and d, are dimensionless and control the sharpness of the quadratic-nonlinearity, the timescale of spike adaptation, the sensitivity of spike adaptation to sub-threshold oscillations, and the magnitude of the spike reset adaptation variable. Crucially Izhikevich and colleagues [36,90], fit parameters for a large class of cortical and sub-cortical neurons, thus affording our model a high degree of neurobiological plausibility while greatly reducing the number of free parameters.

The apical compartment of the L5_PT neuron consists of a two dimensional non-linear system introduced by [37] as a phenomenological model of the Ca²⁺ plateau potential in the apical dendrites of L5_PT neurons.

(3)

(4)

Where describes the regenerative non-linearity underlying the Ca²⁺ plateau potential, and denotes a square wave function of unitary amplitude describing the backpropagating action potential (delayed by 0.5 ms and lasts for 2 ms) with denoting the somatic compartment spike time. The parameters l, g m, denote the leak conductance (nS), amplitude of regenerative non-linearity (pA), and amplitude of the back propagating action potentials (pA) respectively. The model and parameters were derived from a more complex model of Ca²⁺ spikes built to predict in vitro L5_PT spike times [42].

To simulate key observations from empirical experiments, we coupled the compartments together so that sodium spikes in the somatic compartment triggered a back propagating action potential affecting the apical compartment through the square wave function . In turn, plateau potentials in the apical compartment controlled the reset conditions of the somatic compartment. We leveraged the insight [34,40,41] that the difference between regular spiking and intrinsic bursting can be modelled by changing the reset conditions of equations [1] and [2], raising the reset voltage (increasing c) taking the neuron closer to threshold, and reducing the magnitude of spike adaptation (decreasing d). Whenever the membrane potential in the apical compartment exceeded mv the reset conditions changed from regular spiking to bursting parameters. This allowed us to reproduce the transient change in dynamical regime in L5_PT cells that occurs when they receive coincident apical and basal drive. Parameters values for each neuron/compartment are given in Table 1.

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Table 1. Parameters for each neuron L5_PT apical dendrite parameters were taken from [37]. L5_PT soma parameters were modified from the model of intrinsic bursting described in [36], p.290). Basket cell (fast spiking interneuron), and matrix thalamus parameters were taken from [90].

https://doi.org/10.1371/journal.pcbi.1012951.t001

Based on the finding that communication between apical dendrites and the soma of L5_PT cells requires depolarising input from the matrix thalamus to the “apical coupling zone” in L5a [43] we made back propagating action potentials and Ca²⁺ driven parameter switches depend stochastically upon a phenomenological model of excitatory dynamics in the apical coupling zone described by the saturating linear system shown in equation [5].

(5)

Coupling was driven by thalamic spikes (where denotes the time that the thalamic neuron passes the threshold ) and the decay constant was taken from work estimating the decay of the post synaptic excitatory effects of metabotropic glutamate receptors [91] which have been shown empirically to mediate inter-compartmental coupling in L5_PT cells [43]. By design, the dynamics of the coupling variable varied between 0 and 1 and governed the probability with which back propagating action potentials would reach the apical compartment and the probability with which a Ca²⁺ spike would lead to a switch in the soma reset parameters.

Based on previous spiking neural network models of rivalry [25,26,45] the cortical component of the network had a one-dimensional ring architecture. Each point on the ring represents an orientation preference with one full rotation around the ring corresponding to a 180^° visual rotation. This mirrors the fact that a 180^° rotation of a grating results in a visually identical stimulus and also ensures periodic boundary conditions. The cortical ring contained 90 L5_PT neurons and 90 fast spiking interneurons. Each pair of excitatory and inhibitory neurons was assigned to an equidistant location on the ring (unit circle) giving each neuron a 2^° difference in orientation preference relative to each of its neighbours. The (dimensionless) synaptic weights connecting neurons (, , and ), were all-to-all with amplitude decaying as a function of the Euclidean distance between neurons (equation [6]) according to a spatial Gaussian footprint (equation [7]).

(6)

(7)

Where θ is the location of the neuron on the unit circle, κ and ω denote the pre- and post-synaptic neuron type (i.e. ), controls the magnitude of the synaptic weights, and the spatial spread. In line with empirical constraints inhibitory coupling had a larger spatial spread than excitatory to excitatory coupling [92]. Each thalamic neuron received input from 9 cortical neurons and then projected back up to the apical dendrites of the same 9 cortical neurons recapitulating the diffuse projections of higher-order thalamus onto the apical dendrites of L5_PT neurons in layer 1 [47,48]. For simplicity we set projections to and from the thalamus to a constant value (e.g., ). For the sake of computational efficiency we also neglected differences in rise time between receptor types which allowed us to model receptor dynamics with a first-order linear differential (equation [8]) with decay () constants chosen to recapitulate the dynamics of inhibitory (GABA_A), and excitatory (AMPA and NMDA) synapses [93,94].

(8)

Where, as above, denotes the time that the neuron passes the threshold The conductance term entered into the input through the relation where is the reverse potential of the synapse. Following [90] we set to 1 for GABA_A and AMPA synapses, and for NMDA synapses. To prevent artificial distortions of the spike shape that can occur during parameter sweeps that push the model outside its normal operating regime we clipped individual NMDA conductances to a maximum value of 85 nS.

For the threshold detection simulations, the somatic compartment of each L5_PT cell received 600 Hz of independent (Poisson) external drive and apical compartments received 50 Hz of external drive. The whisker deflection was simulated by a pulse of constant amplitude varying between 0 – 350 pA lasting 200 ms and weighted by the spatial Gaussian shown in equation [9] where N is the neuron at the centre of the pulse and the spatial spread.

(9)

For the visual rivalry simulations, separate monocular inputs targeting the somatic compartment of L5_PT cells were modelled with two independent Poisson processes (representing input from the left and right eyes in the case of binocular rivalry or left and right movement selective populations in the case of plaid perception) with rates varying between 1200 and 1800 Hz depending on the simulation. The external drive was weighted by the spatial Gaussian shown in equation [10] centred on neurons 90^° apart on the ring abstractly corresponding to the orthogonal grating stimuli commonly employed in binocular rivalry experiments.

(10)

Here i denotes the index of the i th neuron, and control the orientation of the stimulus delivered to the left and right eyes, and the spatial spread.

To capture the slow hyperpolarising current that traditionally governs switching dynamics in models of bistable perception [53], the somatic compartment of each L5_PT cell was coupled to a phenomenological model of slow hyperpolarising Ca²⁺ mediated K⁺ currents [52] which entered into the external drive term for each cell (i.e. ) with dynamics given by equation [10].

(11)

Where denotes the contribution of each spike to the hyperpolarising current, and the decay constant.

Rather than fit the parameters of our model to individual experimental findings, which permits substantial degrees of freedom and risks overinterpretation of idiosyncratic aspects of individual experiments, we instead elected to challenge a single model to qualitatively reproduce a wide array of experimental findings with a minimal set of parameter changes carefully chosen to reflect experimental manipulations and perturbations. Specifically, we initialised the connectivity parameters such that: 1) when the model received a background drive the conductances were approximately E/I balanced with a coefficient of variation > 1, corresponding to an asynchronous irregular regime [49]; 2) inhibitory connections on the cortical ring had broader (Gaussian) connectivity than excitatory connections generating a winner-take-all regime when the model received two “competing” inputs to opposite sides of the cortical ring; and 3) a slow hyperpolarising current was added to the somatic compartment of each L5_PT cell destabilizing the winner-take-all attractor states leading to spontaneous switches between transiently stable persistent states with an average duration in the experimentally observed range for binocular rivalry. Parameters for the model components described by equations [5] – [11] are supplied in Table 2.

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Table 2. Parameter description, values and units for the model components described by equations [5–11].

https://doi.org/10.1371/journal.pcbi.1012951.t002

The equations were integrated numerically in MATLAB 2023b. The apical compartment was integrated with a standard forward Euler scheme. All other compartments were integrated using the hybrid scheme for conductance based models introduced by [95]. All simulations used a step size of 0.1 ms and were run for 30 s. Unless stated otherwise, all simulation results were averaged over a minimum of 30 random seeds.

Distance to bifurcation

To obtain a closed form expression for the distance to bifurcation in the apical compartment (equations [3-4]), we leveraged the fact that saddle node bifurcations occur when the nullclines ( of the system intersect tangentially [58]. That is, the nullclines and derivative of the nullclines must be equal leading to the following two requirements (where we have absorbed the term describing back propagating action potentials into the external drive which we treat as a constant).

(12)

(13)

We used equation [13] to solve for giving = We then substituted back into equation [12] to solve for yielding the value of the external current at each of the two bifurcations I_B1 = 538.911 pA and I_B2 = 647.375 pA (corresponding to points at which the linear adaptation current nullcline intersects tangentially with the left and right knees of the cubic membrane potential nullcline see Fig AB-AD in S1 Appendix).

Psychometric and neurometric functions

To obtain a measure of response probability from our model comparable to the psychometric functions in Takahashi and colleagues [12,13] we took a two pronged approach. First, for all stimulus intensities including stimulus absent trials (when the model only received a background drive) we calculated the frequency with which the spike count in the 1000 ms post stimulus window exceeded a criterion defined on the interval between the minium and maximium spike count. We then selected the (optimal) criterion that best minimised misses and false alarms. Trials exceeding the optimal criterion were counted as a response. Following Takahashi and colleagues [12,13], we then fit logistic functions (equation [14]) to the network responses using non-linear least-squares.

(14)

Where P(x) is the detection probability (i.e. the probability of the model producing a hit or false alarm), and are free parameters. We used the optimal criterion found in the unperturbed (i.e. control) simulations in the perturbation simulations.

Second, to ensure that our results were not an artefact of the (optimal) criterion we constructed neurometric functions following the procedure described in the supplementary material of (13). Specifically, for each stimulus intensity we constructed ROC curves and then computed the AUC (area under the ROC curve) thereby summing over all criterions. To convert the AUC into a quantity comparable to a psychometric function (i.e. so that each neurometric function vairied between 0 and 1), we normalised the AUC values, . The intercept was given by the minimium AUC across all conditions, and the was given by the maximium AUC across all conditions.