Overview and rationale of network model

There exists inter-species, regional and laminar variations in the cellular composition and connectivity of cerebral cortex. For instance, the proportion of inhibitory interneurons to excitatory pyramidal cells may range from 9% to 20% across layers of mouse somatosensory cortex, and the proportion of PV to non-PV interneuron subtypes from 30% to 60%[3,26]. Yet despite this heterogeneity, there exist conserved features that suggest some differences represent a `variation on a theme’ from which common principles can be understood[27]. For example, PV and SST interneurons demonstrate connectivity preference with the peri-somatic and dendritic regions of pyramidal cells, respectively, and connection probability between pyramidal cells and inhibitory interneurons is typically greater than recurrent connectivity between pyramidal cells [26,28,29]. For the purposes of this study, we consider PV and SST interneurons equivalent to FS and NFS, respectively.

To explore the role of interneuron subtypes upon network activity we developed a model that retained broad themes of cortical organisation with respect to intrinsic neuron electrophysiological features, the proportion of neuron subtypes, connection probability and synaptic properties (code available at https://osf.io/2sr6y/). We assume that incorporating these themes will provide insight into principles of cortical network activity, while acknowledging that our model is not necessarily representative of a specific cortical layer or brain region.

Pyramidal cell model

Excitatory pyramidal cell (PC) models possess both a proximal (‘somatic’) and distal (‘dendritic’) compartment and were optimised to exhibit realistic electrophysiological properties in response to constant current stimuli. The soma contained a rapid-activating sodium (NaT), persistent potassium (K_P), hyperpolarisation-activated (Ih) and leak (L) conductance. The dendrite contained an Ih, leak and high-voltage activated calcium (CaH) conductance, and calcium accumulation/decay kinetics (channel mechanisms of Markram et al.[30] and their cited references). The current-balance equations for somatic (S) and dendritic (D) compartments are (1)

Where C is the membrane capacitance, G the peak channel conductance, v the transmembrane voltage, E the channel reversal potential, R_SD the intracellular resistance between somatic and dendritic compartments, m/h the activation/inactivation gating variables, m_∞(v) and h_∞(v) the steady-state and time-constant functions, and x represents the channel mechanism,. Peak channel conductance, membrane capacitance, intracellular resistance and calcium kinetics were optimised using a feature-based evolutionary algorithm in response to the following external current stimuli based on Hay et al.[31,32]:

1. 5ms, 1.9nA, constant current at the soma
2. 5ms, 1.9nA, constant current & 5ms 0.2nA constant current at the dendrite at offset of somatic stimulus
3. 500ms, 0.5nA, constant current at the soma
4. 500ms, 0.8nA, constant current at the soma

Parameters were optimised to reproduce AP waveform, back-propagating AP (BAP), and BAP-activated Ca²⁺ spike features based on Hay et al[32]. A list of features and objective values is provided in S1 Table. Given that our model contained only two compartments, feature 9 from Hay et al. (bAP amplitude at 620um) was excluded and to ensure realistic time lag between APs generated at the soma and dendrite two additional features were added: times to first spike at the soma and dendrite. Finally, to ensure realistic input-spike frequency characteristics, the model was optimised to exhibit spike frequencies of 10 Hz and 20 Hz to stimuli 3 and 4, respectively, similar to the spike frequency generated by the model of Hay et al. [32]

Interneuron & simplified PC model

FS and NFS interneurons, and the simplified PC model, were described using the formulation of Izhikevich et al[33,34], (2)

Parameter values for FS and NFS interneurons were set to those found in Izhikevich et al[33,34] to replicate features of FS and ‘low threshold spiking’ electrophysiologic subtypes, respectively. The simplified PC model (the “Izhikevich PC model”) exhibited ‘regular spiking’ characteristics. To modify interneuron rheobase a constant current (I) was applied to the soma. Increased rheobase (reduced excitability) was quantified as the percentage rheobase increase relative to baseline in response to a constant depolarising stimulus.

Synapse model

Synaptic conductance dynamics were modelled with a bi-exponential formalism, (3) (4) where t_k denotes spike times of presynaptic neurons, F is a normalisation factor such that peak conductance equals 1uS and W is synaptic weight. Excitatory synapses contain fast (AMPA) and slow (NMDA) conductances with AMPA:NMDA ratio of 0.2, reversal potential of 0mV and τ₁/τ₂ set to 0.1/1.5 ms for AMPA and 2/20 ms for NMDA, respectively[35–37]. Rise/decay inhibitory (GABA_A) conductance time constants were set to 0.25/5 ms with reversal potential of –80 mV.

Synaptic weights & latencies

A range of synaptic post-synaptic potential (PSP) amplitudes and latencies have been recorded from cortex with significant variation across species and brain regions[3,26,28,38–42]. Nevertheless, there exist broad differences between neuronal populations that we have aimed to capture using a parsimonious approach with the following considerations:

1. PSP amplitudes have consistently been observed to be of greatest magnitude from PC to FS neurons and of smallest magnitude between PC neurons.[26,28,38–41,43]
2. Synaptic latencies have consistently been found to be shortest between PC and FS interneurons and longest between PC neurons.[3,26,28,39,40,42]

Given these observations, synaptic weights were adjusted to generate mean peak PSPs of 0.8 mV for PC-to-FS connections, 0.4 mV for PC-to-PC connections and 0.6 mV for other connections. These values are comparable to recordings from mouse somatosensory cortex[26,28]. The distribution of experimentally measured PSPs are typically skewed and so weights were drawn from a log-normal distribution that generated a PSP sample variance of 0.1 mV[26,39]. To account for filtering of synaptic input, synaptic weights onto the PC dendrite were scaled to generate a PSP amplitude two-thirds of the PSP elicited at the soma, a value that approximates the somatic PSP amplitude in response to synaptic input ~150 um from the soma in the PC model of Hay et al. [32]

To reflect population differences in synaptic latencies, mean latencies from PC-to-FS and from PC-to-PC neurons were set to 1.0 ms and 1.8 ms, respectively. Other mean latencies were set to 1.5 ms, values comparable to recordings from mouse somatosensory cortex[26,28,40]. Latencies were drawn from a normal distribution with variance of 0.1 ms[39]. To prevent instantaneous or negative values, PC-to-FS latencies were drawn from a truncated normal distribution with lower limit of 0.1 ms.

Cell numbers & connectivity

The network contained 1800 (80%) PC neurons and 450 (20%) inhibitory interneurons. These values approximate neuron numbers and excitatory-inhibitory cell ratios within a single cortical column layer[26]. Inhibitory interneurons were divided equally between FS and NFS subtypes, which approximates proportions measured in some studies, although variation exists across layers[3,28,30].

Connection probabilities from 5% to 20% have been estimated between PC neurons within a cortical column. In comparison, connectivity between PC and FS/NFS interneurons has been found to be denser but with a wide range of measured values from 10% to over 50%, which may reflect differences between species, brain region and experimental technique[28,29,40,41,44,45]. A number of studies have found denser connectivity between PC and FS compared to PC and NFS interneurons[28,40]. To reflect these trends, we assigned a connection probability of 15% between PC neurons, 25% between PC and FS interneurons and 20% for all other connections. FS interneurons preferentially synapse onto the axon, soma and proximal dendrites of pyramidal neurons in contrast to NFS interneurons which tend to synapse distally[46,47]. To preserve differences in synaptic distribution we assigned 80% of FS/NFS synapses onto the proximal/distal PC compartment. PC-to-PC synapses were more evenly distributed, with 60% assigned to the soma[48].

External stimulation & network simulation

Neurons in each population were stimulated with external Poisson-distributed excitatory and inhibitory synaptic stimuli. Excitatory and inhibitory stimulation frequencies and conductance magnitude for each population were equal to generate `balanced’ external input[49]. Cortical excitatory neurons typically fire sparsely compared to inhibitory interneurons, with mean rates often under 1Hz[50–52]. To provide a baseline from which to assess the relative impact of FS and NFS excitability upon network activity, a grid parameter search of external stimulation frequencies upon each population was performed to elicit equal mean firing rates within the FS and NFS populations of 2.0 Hz, and 0.3 Hz within the PC population. An identical approach was used to generate baseline firing frequencies for the network receiving only excitatory external stimulation. Network simulations were performed with NetPyNE using a fixed integration time step (dt = 0.025ms)[53]. Mean values of network properties were obtained by performing 10 simulations initiated with a different random seed.

One & two-population networks

To isolate the role of interneuron properties and external input upon PC gain, we developed a two-population network with PC and FS interneurons (the “PC-FS network”), and one-population networks with PC, FS and NFS neurons alone (the “FS interneuron network” and “NFS interneuron network”). FS neurons comprised 20% of the population in the PC-FS network (450). Otherwise, cell numbers were identical to the original network. Each network was stimulated by excitatory and inhibitory synapses to elicit the same firing rates as the three-population network under baseline conditions.

Network spike characteristics & excitation-inhibition balance

Inter-spike interval coefficient of variation (ISICV) for neuron n was defined as the ratio of ISI mean (〈ISI_n〉) and ISI standard deviation (σ_ISI,n), (5)

The spike correlation coefficient (CC) was estimated by converting spike trains for each PC neuron into 10 ms binned spike count vectors (x) and calculating the mean cross-correlation between all distinct neuron pairs (a, b) within the PC population (size N), (6)

Values for mean and standard deviation were calculated across ten simulations initiated with a different random seed for external stimulation. To estimate population synchrony, a Fourier transform was performed on the vector of mean population firing rates binned into 10 ms intervals. Results shown are normalised to the peak value at baseline.

The mean conductance from synaptic inputs from population P onto PC neuron n () was calculated as the temporal average of the vector of mean synaptic input conductance onto neuron n (), (7) where i represents individual synaptic inputs from population P and S_n,P is the total number of synapses onto neuron n from population P. P represents either the PC, FS or NFS populations, or input from external stimulation sources. To calculate temporal EI balance the normalised cross-correlation function of the vector of mean conductance onto a PC neuron n from PC () and either FS or NFS () synaptic inputs was first computed, (8) where T is simulation length, k is temporal lag, I is the inhibitory (FS or NFS) population, is the temporal average of the vector of mean synaptic input conductance from the PC population onto PC neuron n and σ_g is the standard deviation of the vector of mean synaptic input conductance onto PC neuron n. To estimate average temporal balance (), 500 PC neurons were selected at random and the mean cross-correlation function of all neurons in this sample was calculated, (9)

Temporal EI balance between PC and either the FS or NFS population was calculated using the same approach.

Assessment of ISN properties & PC gain

In networks containing multiple inhibitory populations, the existence of an ISN can be evaluated by stimulating the inhibitory population and observing a paradoxical reduction of inhibitory current onto the excitatory population in the presence of lower excitatory firing rates [54]. Therefore, we tested for an ISN by applying constant-current stimuli of increasing magnitude to the FS and NFS populations and calculating the change in PC firing rate and inhibitory input current. A stimulus range that elicited up to a 1Hz change in FS and NFS firing rates was used.

To estimate effective synaptic weight of the PC subnetwork, a 1 ms constant current stimulus of 0.9 nA was applied to 20% of PC neurons. An average population response was obtained by applying the stimulus 100 times and calculating the mean spike frequency of the PC population. The stimulus produced a bi-modal response: the first peak from spikes elicited by the stimulus and the second peak via PC-to-PC interactions. To estimate effective synaptic weight, the ratio of spikes recruited via PC-to-PC interactions (P₂) to spikes elicited by the stimulus (P₁) was calculated. Periods corresponding to P₁ and P₂ were obtained from the bi-modal average population response. P₁ was defined from stimulus onset to the minimum between peaks. P₂ was defined from the minimum between peaks to the time at which the average population response returned to this value.

To explore the role of correlated inputs upon PC neuron excitability, artificial spike trains of increasing pairwise spike cross-correlation were generated. Spike correlations were introduced by first creating 100 independent Poisson spike trains of fixed rate, and then shifting randomly selected spikes within a train to fall within 5ms of a spike within a different train. This process was repeated for spike trains of different mean rate.

Optimisation of firing rate model

A three-population Wilson-Cowan rate model was developed to explore changes in network dynamics associated with increased FS and NFS rheobase, (10) where R_x are the mean firing rates of the PC (x = E), FS (x = F) or NFS (x = N) populations, W_yx are the synaptic weights from population x to y, τ_X time constant of the population firing rate responses, B_x are the population gains and I_x are external stimulation currents. Under the baseline condition, gains for all populations were set to 1.

The synaptic weights of the rate model were optimised to reproduce the firing rates of the spiking network under baseline conditions using the following approach. First, to reflect features of the spiking network, inhibitory synaptic weights onto each population were expressed as a proportion of the integral of the PSP elicited by an excitatory synapse. For example, the FS-to-PC weight is given by, (11) where PSP_EF is the post-synaptic potential elicited by an FS synapse onto a PC neuron at the resting membrane potential. This approach was used to calculate W_EN in terms of W_EE, and weights received by the FS & NFS populations in terms of W_FE & W_NE, respectively.

Second, to estimate external stimulation (I_x), the spiking network was simulated under baseline conditions with weights of internal network connections set to zero. I_E, I_F and I_N were assigned mean firing rates of the PC, FS and NFS populations, respectively. Since this approach will contain error related to changes in membrane conductance after removing internal connections, both external stimulation (I_x) and free weight parameters (W_EE, W_FE & W_NE) were optimised using a least-squares algorithm and an iterative approach so that firing rates of the rate model approximated firing rates of the spiking network. First, synaptic weights were optimised so that the rate model firing rates approximated baseline firing rates of the spiking network for fixed I_x. I_x bounds were then relaxed by 10% of their original value, synaptic weights fixed to their updated value, and the optimisation repeated. These steps were repeated until the least-squares error was under 0.001. Final parameter values were: W_EE = 0.95, W_FE = 9.36, W_NE = 9.71, I_E = 0.89, I_F = 1.44, I_N = 2.5.

Estimation of gain with modulation of FS & NFS excitability

To estimate the impact of reductions in FS and NFS excitability upon network dynamics, we assume that synaptic weights and external stimulation remain fixed at baseline values, but the gain (B_x) of each population varies with modulation of interneuron excitability. Therefore, to determine excitatory population gain corresponding reductions of FS excitability in the spiking network, B_E was re-calculated using the optimised synaptic weights and the steady-state firing rates of the spiking network, (12)

The same approach was used to calculate B_E associated with reductions of NFS excitability.

To create a rate model that approximates conditions of reduced FS excitability, PC gain (B_E) was expressed as a function of the total input current onto the PC population (I_T) and fitted with a sigmoidal function, (13)

Therefore, B_E approximates PC population gain as total input current increases due to reductions of FS excitability. This approach was also used to estimate B_E under conditions of reduced NFS excitability to yield two rate models: one for reductions of FS and one for reductions of NFS excitability. Phase portraits and bifurcation analyses were performed using XPPAUT[55].

Statistics

Results and error bars shown in Figures represent mean ± s.e.m. Comparisons of spiking characteristics, EI balance and changes in PC gain for a given FS/NFS rheobase value were performed using Welch’s t test. Changes in PC gain and population characteristics compared to baseline across different rheobase values were performed with one-way ANOVA and post-hoc Tukey’s test. Differences were considered significant if P < 0.01.

Network model

We developed a network model that incorporated connectivity and synaptic parameters that reflect broad differences between FS and NFS interneuron populations observed in experimental data (Fig 1A and 1C). Notably, connection probability was highest between the FS and PC populations and lowest within the PC population, and synaptic latencies shortest between the FS and PC populations (Fig 1). We used a network state in which mean firing rates of the FS and NFS populations are equal (2Hz) as a baseline condition from which to evaluate the impact of interneuron excitability upon network activity.

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Fig 1. Characteristics of network model.

A) The network contained one excitatory (PC) and two inhibitory (FS and NFS) populations stimulated by external (E_ext and I_ext) input. Neuron models possessed intrinsic electrophysiological properties typical for each population and, under the baseline conditions, both the FS and NFS populations had mean firing rates of ~2Hz. B) Raster plot of network activity under baseline conditions. C) Distributions of synaptic latencies and post-synaptic potentials for all synaptic connections onto the PC population. FS to PC connections had shorter latencies and elicited greater post-synaptic potential amplitudes compared with NFS to PC connections.

https://doi.org/10.1371/journal.pcbi.1009521.g001

Emergent network properties: Spike characteristics & EI balance

A remarkable feature of cortical networks is their ability to maintain a precise balance of excitation and inhibition under different behavioural conditions[12,56,57]. Certain characteristics of EI balance, such as the net current (i.e., the sum of excitatory and inhibitory currents) and temporal relationship determine cortical spiking characteristics and computational properties[58]. Therefore, we evaluated the spiking characteristics and EI balance of our network during the baseline condition to provide a point of comparison with experimental studies.

Under the baseline condition, spiking was highly irregular with a mean ISI coefficient of variation (ISICV) of 1.1 and a long-tailed distribution of inter-spike intervals, qualitatively similar to in-vivo observations (Fig 2, spike raster plot shown in Fig 1)[59,60]. The mean firing rate of most PC neurons (77%) was under 0.5 Hz with a range from 0 to 1.9 Hz and spike correlations within the PC population were very small (0.001) consistent with an irregular and asynchronous network regime[8,61].

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Fig 2. Network spike characteristics and excitation-inhibition balance.

A) Distributions of spike characteristics for PC neurons under baseline conditions. PC spiking is irregular with a long-tailed ISI distribution and mean ISICV of 1.1. B) Breakdown of excitation-inhibition balance and synaptic input current sources onto the PC population during the baseline condition. Points within each cloud represent mean input current from different sources onto individual PC neurons. Mean current from external stimulation (E_ext, shown in orange) represented the majority (97%) of mean total excitatory current (E, shown in yellow). A similar proportion of inhibitory current was derived from both external (I_ext) and network sources (FS & NFS). Net current (E+I) was of similar magnitude to total excitatory and inhibitory currents (denoted E & I, respectively). Net conductance onto the PC population (S1 Fig) was dominated by inhibitory inputs. C) Mean cross-correlation between synaptic conductance’s received from the PC and FS (PC-FS), NFS (PC-NFS) or both inhibitory populations (PC-I) onto the PC population. There is stronger and more rapid cross-correlation between the PC and FS, compared to PC and NFS, populations (peak cross-correlation at 6 and 12ms, respectively). The temporal lag and co-fluctuation of PC and FS synaptic conductance’s can be appreciated from a trace of mean synaptic conductance onto the PC population (C, lower).

https://doi.org/10.1371/journal.pcbi.1009521.g002

To quantify EI balance, we recorded mean synaptic currents and conductance from excitatory and inhibitory inputs onto each PC neurons. Most excitatory current was obtained via external stimulation, and only 3% of the mean total excitatory current onto the entire PC population was obtained via internal network activity from other PC neurons (Fig 2). In contrast, 51% of mean total inhibitory synaptic current was obtained from the FS and NFS populations (32% and 19%, respectively). Since FS and NFS populations fire at similar rates under baseline conditions, greater current from the FS population is due to denser connectivity from FS to PC neurons. Finally, mean net current was of similar magnitude to total excitatory current (-0.23 nA compared to -0.43 and 0.2 nA, respectively, Fig 2) and net conductance was inhibition-dominated with an EI conductance ratio of 0.5 (S1 Fig). The presence of inhibition-dominated synaptic input onto the PC population is consistent with previous intracellular cortical recordings, and the magnitude of net current onto the PC population suggests a ‘loosely balanced’ network regime[49,58]. Similar EI characteristics were observed within the FS and NFS populations (S1 Fig).

Intracellular recordings from cortical pyramidal cells have revealed the existence of fast temporal correlations between excitatory and inhibitory synaptic currents, suggesting inhibitory neurons can be rapidly recruited by moment-to-moment fluctuations of excitatory activity[10]. Temporal EI balance may play an important role in maintaining asynchronous activity within densely connected cortical networks[7]. We investigated temporal EI balance characteristics by calculating the mean cross-correlation between network-driven excitatory and inhibitory synaptic conductances onto the PC population. We observed a peak in excitatory-inhibitory cross-correlation onto the PC population at a short time lag of 6ms, consistent with rapid activation of the inhibitory population in response to PC spiking (Fig 2). To dissect the relative contributions of FS and NFS interneurons to temporal EI balance, we repeated this analysis for synaptic conductances from each interneuron population. Stronger correlation between PC to FS compared with PC to NFS conductance was observed (0.07 and 0.019, respectively), and peak PC to FS correlations occurred at a shorter time lag compared to PC to NFS (6 ms and 12 ms respectively, Fig 2). Correlated excitatory-inhibitory synaptic input is appreciated visually with a trace of normalised mean excitatory and inhibitory conductance onto the PC population (Fig 2). Our model, therefore, exhibits rapid recruitment of inhibition in response to fluctuations of excitatory activity, with FS neurons providing faster and more robust inhibitory feedback.

Modulation of network activity through FS and NFS interneurons

There is accumulating evidence from studies employing targeted optogenetic silencing or activation that interneuron subtypes may exert selective influence over cortical network state[62–65]. This idea is supported by the observation that some neurotransmitters selectively modulate the excitability of interneuron subtypes[23–25]. Therefore, we explored the impact of reductions of FS and NFS excitability upon network spike characteristics and EI balance. Interneuron excitability was modulated by increasing the rheobase (or spiking threshold) of neurons within the FS and NFS populations (Fig 3).

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Fig 3. Changes in network properties with modulation of interneuron excitability.

A) FS interneuron input-frequency relationship following modulation of interneuron excitability in response to random excitatory synaptic input. Interneuron excitability was quantified as the percentage increase in rheobase relative to baseline in response to a constant current stimulus (shown here for baseline, 25% and 50% increase in rheobase). B) Increased NFS rheobase produced stepwise reductions of NFS firing rates and increases of FS & PC firing rates. Increased NFS rheobase did not produce a significant change in PC spike correlations (C) or population oscillations (D). In contrast, an increase in FS firing rates beyond a 25% increase in FS rheobase was observed (B, right), together with large increases in both PC spike correlations and population synchrony (normalised relative to baseline, C & D). E) Changes in characteristics of excitatory and inhibitory input current onto the PC population with modulation of interneuron excitability. As FS rheobase was increased, the proportion of excitatory current onto the PC population derived from within the network (i.e., from other PC neurons, denoted Int) increased from ~3 to 9% (E, upper left). Since external stimulation remained fixed, this change in network-derived excitatory current must drive increased firing rates observed in B. Furthermore, increased FS rheobase also produced greater network-derived inhibition (E, top right). Despite increased population firing rates, a paradoxical reduction of EI balance was observed (E, bottom).

https://doi.org/10.1371/journal.pcbi.1009521.g003

Increasing NFS rheobase reduced NFS firing rates, as would intuitively be expected, and increased PC and FS firing rates consistent with disinhibition from the NFS population (Fig 3). As NFS firing rates approach zero PC and FS firing rates plateaued at ~0.4 Hz and ~3.2 Hz, respectively (S1 Fig). These changes suggest that the PC population exerts only small influence over NFS firing rates, relative to external excitatory stimulation, because NFS firing rates drop despite higher PC firing rates and excitatory input. Furthermore, the FS population appears to adopt a compensatory role to provide sufficient inhibition to maintain an upper limit of PC firing rates. Finally, a significant change in PC spike correlations and population synchrony was not observed under these conditions (Fig 3).

Increased FS rheobase produced a qualitatively different influence upon network activity. Here, the FS population firing rate exhibited a U-shaped relationship when rheobase was increased beyond 25% (Fig 3). PC and NFS firing rates also increased. This transition in network activity was associated with a significant and greater than 10-fold increase in both spike correlations (from 0.001 to 0.012, P < 0.001 for FS rheobase ≥ 10%, Welch’s t test, Fig 3) and oscillatory power between 20 and 25Hz within the PC population (P < 0.001 for FS rheobase ≥ 10%, Welch’s t test, Fig 3). Stronger temporal EI correlation between excitatory and inhibitory synaptic inputs were also observed (S1 Fig). Interestingly, despite higher firing rates within the PC population, EI balance within the network transitioned to a more inhibitory state with a reduction of the EI ratio and greater recruitment of network-driven inhibitory activity (Fig 3). Since external stimulation was fixed across all conditions, increased FS firing rates are therefore driven by increased internal excitatory network activity via the PC population (Fig 3). Likewise, an increase in PC firing rates despite higher inhibitory interneuron firing rates implies the existence of enhanced recurrent excitation within the PC population.

Fast spiking interneurons promote the transition into an inhibition-stabilised state

The presence of strong coupling between excitatory neurons can endow cortical networks with unique computational properties, such as input amplification and pattern completion associated with memory formation, but are also at risk of developing pathologic excitatory activity unless stabilised by recurrent inhibition[15]. Networks with these properties, known as inhibition-stabilised networks (ISN), have been confirmed experimentally by their paradoxical response to external stimulation of the inhibitory population: a reduction of steady-state firing rate in both the inhibitory and excitatory populations[16,66]. After increasing the rheobase of the FS population, we observed changes in network activity to suggest strong coupling between PC neurons and hypothesised that these changes may represent the transition into an ISN. Therefore, we tested for the presence of an ISN by applying an external stimulus to both FS and NFS populations (Figs 4 and S2). Since our network contains two inhibitory populations an ISN is characterised by a reduction in the magnitude of mean inhibitory current onto the PC population despite the presence of lower PC firing rates[54].

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Fig 4. Reduced FS excitability promotes a paradoxical response to inhibitory stimulation.

A) The presence of an ISN was explored by applying an external current stimulus to the FS (y axis) and NFS (x axis) population and recording the mean change in inhibitory (hyperpolarising) current received by the PC population (increased inhibitory current in green, decreased inhibitory current in red). This process was repeated after increasing FS (B & C) and NFS (S2 Fig) rheobase. During the baseline condition and increased NFS rheobase, external stimulation of the inhibitory population increased inhibitory current received by the PC population. In contrast, increased FS rheobase (C) was associated with a paradoxical reduction of inhibitory current received by the PC population despite lower PC firing rates, consistent with an ISN regime. This effect was more pronounced with greater increases of FS rheobase (panel 3).

https://doi.org/10.1371/journal.pcbi.1009521.g004

Under the baseline condition, external stimulation of the FS and NFS populations enhanced net inhibitory current onto the PC population and reduced PC firing rates, consistent with a non-ISN (Fig 4). These changes suggest that greater activity within the inhibitory population exerts a direct inhibitory influence over the PC population, and that reductions of PC firing rate have little bearing over the recruitment of inhibitory neurons. We observed similar findings under conditions of reduced NFS excitability, suggesting that modulation of NFS excitability does not alter the network state (S2 Fig).

With reductions of FS excitability under 25%, external stimulation of the inhibitory population also revealed changes consistent with a non-ISN (Fig 4B and 4C). However, further suppression of excitability produced a paradoxical reduction in the magnitude of inhibitory current onto the PC population despite lower PC firing rates, consistent with an ISN (Fig 4). Since more external current is applied to the inhibitory population, this ‘reduction of inhibition’ implies diminished recruitment of feedback inhibition driven by the PC population. A more pronounced paradoxical response was observed with greater increases of FS rheobase.

Overall, these results show that interneuron subtypes exert differential control over network state in our model. Notably, reduced FS excitability promoted a regime in which firing rates are driven by PC neurons and are associated with stronger spike correlations and oscillatory activity. To test the generality of these findings, we investigated if network modulation through FS and NFS interneurons is dependent upon either the characteristics of external stimulation applied to the network, or the properties of the model used for the pyramidal cell population. Therefore, we stimulated our network with just excitatory inputs (as opposed to balanced inputs) to reproduce the baseline firing rates, and also developed a second network that contained a simplified single-compartment pyramidal cell model, similar to that used for the FS and NFS populations (the “Izhikevich network”, S3 and S4 Figs). We observed qualitatively similar changes to the original network under both conditions: increasing FS rheobase produced an unexpected reduction of EI balance, the emergence of population oscillations, and promoted a more pronounced paradoxical response to stimulation of the inhibitory populations (S4 Fig). Interestingly, one distinguishing feature in the Izhikevich network is the absence of strong spike correlations associated with increased FS rheobase. This difference in considered further in the Discussion.

Fast spiking interneurons modulate excitatory gain in a rate-based model

Our network model suggests that FS interneurons can preferentially mediate a transition of network activity into an inhibition-stabilised state. To obtain further insight into the dynamics of this transition, we developed a Wilson-Cowan firing rate model with synaptic weights estimated from features of the detailed network (Fig 5). The rate model was first optimised to reproduce firing rates of the detailed network during the baseline condition with the gain of each population (B_X) set to 1. Under the assumption that synaptic weights are fixed, the firing rates during reduced FS and NFS excitability were then reproduced by re-fitting the gain of each population. This resulted in two distinct models that approximate conditions of reduced FS and NFS excitability, respectively.

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Fig 5. FS interneurons enhance PC gain and promote transition to an ISN regime in a rate model.

A) A rate model was optimised with synaptic weights (W) based on features of the spiking network. During baseline conditions the gain of each population was set to 1.0. B) Under the assumption that synaptic weights remain fixed, changes of PC population gain (B_E) in the rate model were required to reproduce firing rates from the spiking network associated with increased FS/NFS rheobase. Increased FS rheobase increased B_E such that the effective recurrent excitatory synaptic strength (B_E x W_EE) exceeded 1.0. C) Two rate models were developed based on changes in PC gain shown in B: one approximating increased NFS rheobase (‘NFS model’) and one FS rheobase (‘FS model’, firing rates during baseline conditions in lower left caption). C) The FS model loses stability via a Hopf bifurcation as B_E x W_EE exceeds 1.0 (asterix) and exhibits sustained oscillations, whereas a qualitative change in network behaviour does not occur in the NFS model. D) Phase portraits and firing rate trajectories of the NFS and FS models for inputs shown in C under the condition of fixed NFS firing rate. The PC nullcline of the FS model has a rightward inflection, and stimulation of the FS population moves the fixed point to lower PC and FS firing rates, suggestive of an ISN regime. In contrast, stimulation of the FS population in the NFS model increases the steady-state FS firing rate, suggestive of a non-ISN regime. These findings are confirmed by changes of inhibitory current onto the PC population in response to stimulation of the FS and NFS population (S5 Fig).

https://doi.org/10.1371/journal.pcbi.1009521.g005

To reproduce the baseline firing rates of the detailed model, the PC-to-PC synaptic weight (W_EE) was optimised to a value of 0.95 and firing rates settled to a stable fixed point (Fig 5). Networks in which the effective excitatory-to-excitatory synaptic weight (B_EW_EE) is greater than 1 can generate self-sustained excitatory activity required for an ISN regime[15]. Consequently, although at baseline the rate model did not behave as an ISN (confirmed by the response of the model to external stimulation of the inhibitory population, S5 Fig), it is possible a small increase in PC gain could reproduce this condition.

To reproduce firing rates corresponding to increased NFS rheobase within the detailed model, the gain of the PC population dropped below 1 (Fig 5). Stimulation of the PC population within the NFS model, which approximates conditions of increased NFS rheobase, did not produce a qualitative change in network behaviour with firing rates again reaching a stable fixed point (Fig 5). Furthermore, consideration of both the FS-PC phase portrait (Fig 5C and 5D) and the response to external stimulation of the inhibitory population reveal features consistent with a non-ISN regime (S5 Fig). Here, external stimulation of the FS population (a rightward translation of the FS nullcline) decreases steady-state firing rates of both the PC and FS populations.

In contrast, to reproduce firing rates corresponding to increased FS rheobase, the gain of the PC population increased such that B_EW_EE exceeds a value of 1 (Fig 5). When stimulating the PC population of the FS model to approximate conditions of increased FS rheobase, the model underwent a Hopf bifurcation and exhibited sustained oscillations (Fig 5). Furthermore, both the FS-PC phase portrait and the model’s response to external inhibitory stimulation demonstrated features consistent with an ISN (Figs 5D and S5). Compared to the NFS model, the PC nullcline has a rightward inflexion (Fig 5). With sufficient external input, the FS nullcline passes above this inflexion, and stimulation of the FS population leads to a reduction in steady-state firing rates of both PC and FS populations. Therefore, to reproduce firing rates of the spiking network, the FS and NFS rate models exert different influence over the gain of the PC population. Notably, reduced FS excitability is associated with increased PC gain that generates characteristics of an inhibition-stabilised state.

Excitatory gain modulation in the detailed network

The findings from our rate models and the existence of a paradoxical response to inhibitory perturbation in the spiking network provide indirect evidence that reduced FS excitability enhances the effective synaptic weight (B_EW_EE) of the PC population. We sought to directly confirm this finding within our spiking network, and to then investigate mechanisms through which B_EW_EE is modulated via the FS interneuron population.

To measure the effective synaptic weight in our detailed network, we perturbed a fraction (20%) of the PC population with a brief current stimulus and calculated the average response of the network (Fig 6). An estimate for B_EW_EE was obtained by measuring the ratio of PC neurons recruited via PC-to-PC interactions compared to spikes directly evoked by the stimulus. According to our hypothesis, if B_EW_EE is less than 1, fewer PC neurons should be recruited through PC-to-PC interactions. Conversely, if B_EW_EE is greater than 1, more PC neurons should be recruited through PC-to-PC interactions, suggestive of excitatory subnetwork instability.

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Fig 6. PC gain modulation in the spiking network.

A) Raster plot of PC spiking (spikes denoted in red) and average population response (top trace, grey) in response to a brief stimulus applied to 20% of the PC population (black arrow). The stimulus elicited a bi-modal population response: the first peak (P₁) in response to the stimulus and the second peak (P₂) from PC-to-PC interactions. The effective synaptic weight was estimated by calculating the ratio of the area under the curves of P₂ (yellow) to P₁ (blue). B) Estimated effective synaptic weight (B_EW_EE) under conditions of increased FS and NFS rheobase. FS rheobase increased B_EW_EE above 1, consistent with excitatory subnetwork instability, and produced a significant increase in B_EW_EE compared to NFS rheobase (P < 0.001). C) Examples of normalised average population response to the perturbation (arrow) following increased FS rheobase.

https://doi.org/10.1371/journal.pcbi.1009521.g006

At baseline, fewer PC neurons were recruited through PC-to-PC interactions, consistent with an estimated value of B_EW_EE under 1 (Fig 6). As FS rheobase is increased, we observed a progressive increase in B_EW_EE that exceeded 1, with more PC neurons recruited through PC-to-PC connections (P < 0.001 for greater than 10% increase in rheobase, one-way ANOVA with post-hoc Tukey’s test). Furthermore, B_EW_EE was significantly higher for increased FS compared to NFS rheobase (P < 0.001, Welch’s t-test, for rheobase values of 10, 20, 30, 40 & 50%). In contrast, increased NFS rheobase did not increase B_EW_EE above 1, suggesting that the excitatory subnetwork remains stable under these conditions. These findings are consistent with results from our simple model and, given that synaptic weights in the model are fixed, provide further evidence that reductions of FS excitability enhance PC gain.

Mechanisms of gain modulation

We next investigated potential mechanisms through which the excitability of the FS population can modulate PC gain. Since we can assume the intrinsic and synaptic properties of the PC model are fixed, we surmise that changes of gain are related to input conditions to which the PC population is exposed. Therefore, we explored the influence of three changes to input conditions associated with increased FS rheobase: reductions of inhibitory current, delayed feedback inhibition, and stronger excitatory input correlations.

First, we note that increased FS rheobase initially reduces total mean inhibitory current onto the PC population (S6 Fig, also appreciated by an increase of EI ratio in Fig 3). A change to the balance of excitatory and inhibitory inputs have previously been demonstrated to enhance neuronal gain, and it is possible reductions of inhibition may exert a similar influence in our network[63,67,68]. Therefore, to test if a reduction of inhibitory input can enhance gain in the absence of input correlations, a network containing only PC neurons (the “PC network”) was created and stimulated with Poisson-distributed inputs that reproduce the conditions of the three-population network (S6 Fig). PC gain was assessed by stimulating a fraction of the PC neurons and recording the average spiking response of the network (S6 Fig). Here, we used a simplified measure of gain: since the PC network contains no inhibitory feedback, a value of B_EW_EE above 1 will generate persistent excitatory activity, which we observed after a ~10% reduction of inhibition (S6 Fig). However, although reductions of inhibitory input caused by increased FS rheobase may contribute to enhanced PC gain, increased NFS rheobase in fact produced greater reductions of inhibitory input yet did not modulate PC gain (S6 Fig). Therefore, there must also exist other mechanisms associated with increased FS rheobase that enhance PC gain.

Mean synaptic latencies between PC and FS interneurons are shorter compared to NFS interneurons in our detailed network (1.0 ms and 1.5 ms, respectively), and it is possible that delayed feedback inhibition may contribute to differential PC gain modulation. To test this hypothesis, a network with a homogeneous interneuron population comprising just FS neurons was developed (the “PC-FS network”). PC and FS parameters were otherwise identical to the original network (Fig 7). To isolate the role of feedback inhibition upon PC gain, we estimated the value of B_EW_EE after progressively increasing mean synaptic latency between the PC and FS population to 1.5 ms (denoted 100% of the NFS value, Fig 7B and 7D) whilst preserving all other synaptic properties. Prolongation of synaptic latency produced a progressive increase in mean estimated B_EW_EE, and B_EW_EE was significantly greater than baseline values (P < 0.01, Welch’s t-test, Fig 7). Therefore, an isolated prolongation of feedback inhibition, similar to that provided by the NFS population, can indeed enhance PC gain relative to the faster feedback inhibition provided by the FS population. We also used the PC-FS network to explore the contribution of differences in synaptic connectivity and distribution between the FS and NFS populations upon PC gain modulation. Both connectivity and distribution also increased B_EW_EE, but only connectivity generated a significant change compared to the baseline value (Fig 7D.

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Fig 7. Interneuron properties and PC gain modulation.

A) A network with a homogeneous FS interneuron population was developed to explore the role of interneuron synaptic properties upon PC gain. B) Histogram of FS-to-PC synaptic latencies before (purple, ‘FS’) and after (green, ‘NFS’) modification to resemble NFS latencies. Connectivity and synaptic distribution were modified in a similar manner. C) Example of normalised average population response to the perturbation (arrow) following modification of interneuron latency. C) Estimated effective PC synaptic weight (B_EW_EE) within the FS network after modifying interneuron properties to resemble the NFS population (100% represents NFS values). An increase in estimated B_EW_EE was observed across all parameters, and at 100% latency and connectivity generated a significant change compared to baseline (P < 0.01, Welch’s t test).

https://doi.org/10.1371/journal.pcbi.1009521.g007

Considering these findings, we wondered if the intrinsic electrophysiological properties of FS and NFS interneurons are also relevant for feedback inhibition and gain modulation. To address this question, we created networks containing only FS or NFS interneurons (the “FS interneuron network” and “NFS interneuron network”, Fig 8). The FS and NFS interneuron networks were then stimulated with excitatory synaptic input designed to replicate fluctuations of input from the PC population (Fig 8A and 8B). Although both the FS and NFS interneuron networks generated similar total spikes to excitatory inputs (S6 Fig) we observed a significantly shorter response latency in the FS interneuron network of over 2 ms (P < 0.001 for input rates above 0.5 Hz, Welch’s t test, Fig 8). Importantly, this value is greater than the difference in mean synaptic latency between the FS and NFS populations that was sufficient to enhance PC gain, implying that rapid intrinsic response properties to synaptic input also afford the FS population influence over PC gain (Fig 7B and 7C).

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Fig 8. Intrinsic interneuron electrophysiological properties.

A) To explore the influence of interneuron electrophysiological properties, a network of just FS or NFS interneurons was developed and stimulated with excitatory synaptic input of increasing mean frequency (‘Stim’). B) Normalised average interneuron population responses to synaptic input demonstrating differences in mean response latency between the FS and NFS populations. C) Mean latency to peak response as a function of excitatory input frequency. Excitatory inputs generated a more rapid-onset response within the FS compared to NFS population (P < 0.001 for input rates above 15 Hz). D) Efficacy of modifying intrinsic electrophysiological and synaptic NFS interneuron properties upon rescuing the original three-population network from an ISN regime. The spiking network was first initialised into an ISN by increasing FS rheobase by 50% (‘Baseline’). Specific properties of the NFS population were then modified to resemble the FS population, and a paradoxical response assessed by perturbing the FS and NFS populations, calculating the change of inhibitory current onto the PC population and then averaging these responses (paradoxical response denoted by negative values in red, e.g., Fig 4). Synaptic connectivity, distribution and latency were adjusted incrementally to resemble the FS population (25, 50, 75 & 100%).

https://doi.org/10.1371/journal.pcbi.1009521.g008

Finally, we explored if the emergence of excitatory spike correlations can also enhance gain. This was achieved by stimulating PC neurons with artificial spike trains containing progressively stronger correlations that approximated those arising within our network during increased FS rheobase (S7 Fig). We found that spike correlations significantly enhanced PC excitability: at an input rate that elicits a mean frequency of 10 Hz for uncorrelated inputs, a spike cross-correlation of 0.0025 elicited a mean frequency of 29.5 Hz (P < 0.001, Welch’s t test, S7 Fig). Furthermore, these findings held regardless of the distribution of spikes across different regions of the PC model, consistent with previous studies demonstrating gain modulation through enhanced fluctuations of membrane potential [69,70]. Indeed, we found that reductions of FS excitability produced a large and significant increase in excitatory current variance onto the PC population (P < 0.001 for FS rheobase ≥ 10%, Welch’s t test, S7C and S7D Fig).

In summary, three mechanisms together enhance gain within the PC population of our network: reduced inhibitory input, delays of feedback inhibition, and excitatory input correlations. An important difference between the FS and NFS population is the rate at which they can provide feedback inhibition in response to excitatory input, a property associated with both synaptic delay and their intrinsic electrophysiological properties. Suppressing FS excitability will preferentially shift feedback inhibition within the network onto the NFS population. Since NFS-mediated inhibition is provided at longer latencies, this in turn may enhance gain within the PC population to promote a transition in network state.