Inhibitory stabilized network behaviour in a balanced neural mass model of a cortical column

Strong inhibitory recurrent connections can reduce the tendency for a neural network to become unstable. This is known as inhibitory stabilization; networks that are unstable in the absence of strong inhibitory feedback because of their unstable excitatory recurrent connections are known as Inhibition Stabilized Networks (ISNs). One of the characteristics of ISNs is their “paradoxical response”, where perturbing the inhibitory neurons with additional excitatory input results in a decrease in their activity after a temporal delay instead of increasing their activity. Here, we develop a model of populations of neurons across different layers of cortex. Within each layer, there is one population of inhibitory neurons and one population of excitatory neurons. The connectivity weights across different populations in the model are derived from a synaptic physiology database provided by the Allen Institute. The model shows a gradient of excitation-inhibition balance across different layers in the cortex, where superficial layers are more inhibitory dominated compared to deeper layers. To investigate the presence of ISNs across different layers, we measured the membrane potentials of neural populations in the model after perturbing inhibitory populations. The results show that layer 2/3 in the model does not operate in the ISN regime but layers 4 and 5 do operate in the ISN regime. These results accord with neurophysiological findings that explored the presence of ISNs across different layers in the cortex. The results show that there may be a systematic macroscopic gradient of inhibitory stabilization across different layers in the cortex that depends on the level of excitation-inhibition balance, and that the strength of the paradoxical response increases as the model moves closer to bifurcation points. Author summary Strong feedback inhibition prevents neural networks from becoming unstable. Inhibition Stabilized Networks (ISNs) have strong inhibitory connections combined with high levels of unstable excitatory recurrent connections. In the absence of strong inhibitory feedback, ISNs become unstable. ISNs demonstrate a paradoxical effect: perturbing inhibitory neurons in an ISN by increasing their excitatory input results in a decrease in their activity after a temporal delay instead of increasing their activity. Here, we developed a neural mass model of a cortical column based on neurophysiological data. The model shows a gradual change in inhibitory stabilization across different layers in the cortex where layer 2/3 is less inhibitory stabilized and shows no paradoxical effect in contrast to layer 4 and layer 5, which operate in the ISN regime and show paradoxical responses to perturbation.

Even though cortical neurons receive a large number of feed-forward connections, mainly 2 originating in the cortex, local recurrent connections heavily modulate their firing 3 patterns [30,31]. Neurophysiological data show that recurrent excitatory connections 4 amplify feed-forward input signals, enabling pattern completion or other computations 5 in the cortex, increasing the speed of network processing or increasing the capacity 6 of memory storage [8,12,18,19,29]. Recurrent excitatory connections can lead to 7 runaway excitation; however, strong feedback inhibition acts to prevent the network from 8 becoming unstable [14,27]. This is known as inhibitory stabilization and networks that 9 combine high levels of excitatory recurrent connections with strong inhibitory feedback 10 are known as Inhibition-Stabilized Networks (ISNs) [37]. 11 Inhibitory stabilization plays an important role in maintaining balance in networks 12 of cortical neurons. In the absence of stabilizing inhibitory feedback, the network may 13 experience runaway activity resembling pathological states, such as epileptic seizures [20, 14 23]. However, impaired inhibitory stabilization does not necessarily result in catastrophic 15 behavior such as seizures; in some cases, it may result in hyperactivity of the neurons 16 or an increase in the cross-correlation between different types of neurons [36]. This 17 highlights the importance of studying the detailed circuitry of balanced networks, the 18 role of strong inhibitory connections in maintaining balance, and the effects of impaired 19 inhibitory stabilization on the activity profiles of neurons. 20 One of the characteristics of ISNs is their paradoxical response where increasing 21 the input to the inhibitory neural population results in a decrease of their activity 22 March 8, 2023 2/34 instead of increasing their activity [32]. This is the signature of ISNs, and several 23 experiments have been conducted to explore the possibility of the existence of ISNs 24 in the cortex [1,3,16,17,33]. The presence of ISNs in the brain had been inferred 25 through internal characteristics of the network of neurons, such as surround suppression 26 or theta oscillations [28,37]. Optogenetic stimulation experiments allow the possibility 27 to investigate the direct effects of perturbing inhibitory neurons [6,13,40] and several 28 experimental studies have shown evidence validating the presence of paradoxical effects 29 in different areas of the cortex in awake mice [1,16,17,33]. 30 However, some experimental studies argue against the presence of ISNs in the cortex 31 or show a mixture of results across different layers in the cortex [3,21]. The study by 32 Atallah et al. [3] showed no paradoxical effect in layer 2/3 of the primary visual cortex 33 of anaesthetized mice and recent experimental work by Mahrach et al. [21] showed a 34 paradoxical effect in layer 5 of the anterior lateral motor cortex (ALM) and barrel cortex 35 but no paradoxical effect in layer 2/3 of the ALM. 36 In this study, a neural mass model (NMM) of a cortical column is developed based 37 on parameters derived from recent anatomical and physiological studies by the Allen 38 Institute [5,34]. This data from the Synaptic Physiology database 1 provides experimen- with neruophysiological data by Markram et al. [22] and Wang [39] who showed that 49 superficial layers in the cortex have more inhibition-dominated activity compared to 50 deeper layers, such as layer 5, that show more excitation-dominated connectivity.

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The structure of the model neurons also receive excitatory and inhibitory input from other motifs in the network, as 61 well as external excitatory inputs from outside of the column (thalamic and inter-cortical). 62 The pattern of connectivity between populations of neurons and the strengths of the 63 excitatory and inhibitory connections are derived from the recently published Synaptic 64 Physiology database (Allen Institute, USA) [5,34]. This data provides estimates of the 65 connection probabilities and synaptic strengths of the different classes of connections 66 within a cortical column (see S1 Table and S2 Table, which are derived from Figure 4A 67 and Figure 4B in [5] weights of inhibitory to excitatory connections in layer 2/3 is given by The connectivity weights for the intracolumnar connections are given in Table 1 where v A i is the average membrane potential of a population of neurons, superscript 92 A indicates either excitatory (E) or inhibitory populations (I), and the subscript i 93  Table 2 [26]. The membrane capacitance is given by c = 0.35 nF. The 96 differences in capacitance of different cell types result in different time constants and 97 so are incorporated within the modelled time constants of the populations. I A syn,i is the 98 weighted summation of the incoming currents, I AB ij , given by where w AB ij are the connectivity weights from population B in layer j to population A 100 in layer i, displayed in Table 1, multiplied by a scaling connectivity constant, k. This 101 scaling factor brings the values that are measured in single cell recordings to the correct 102 operating regime for the neural mass models, while maintaining the relative strengths 103 between them. decaying current [7,9], where τ AB constants are derived from the recent computational study by Billeh et al. [5], who used 109 alpha functions to describe synaptic currents based on electrophysiological studies by 110 Arkhipov et al. [2]. Using MATLAB curve fitting tools (fit), we fit exponential functions 111 to the alpha functions and calculated the resulting single-exponential synaptic time 112 constants, which are given in Table 1 [2,5].

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The function φ() determines the output firing rate of a neural population by applying 114 a sigmoid function to the membrane potential, where φ A max for A=E,I is the maximum firing rate of the neural population, v 0 is the 116 postsynaptic potential for which the firing rate reaches half of its maximum value 117 and r determines the gain (steepness) of the sigmoidal firing rate function.We are 118 making an approximation by choosing a universal output firing rate function for different 119 populations. w E Connectivity weight from other cortical areas/thalamus to excitatory populations 0.1 (inferred from [25] w I Connectivity weight from other cortical areas/thalamus to inhibitory populations 0.2 (inferred from [25] Each population of neurons in the model receives input from other cortical areas 121 or the thalamus. The dynamics of the synaptic equations for the input to excitatory 122 and inhibitory populations, φ A inp for A=E,I, follows the same patterns as other synaptic 123 currents.

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The default values for φ I inp and φ E inp are 1 Hz which are chosen based on experimental data 126 by Arkhipov et al. [2] showing the spontaneous firing rate of excitatory and inhibitory 127 neurons are about 1 Hz. Excitation-inhibition balance is a fundamental characteristic of networks of neurons in 130 vivo [15] and in vitro [35]. The excitation-inhibition balance in a layer of the model 131

Excitation-inhibition balance
is evaluated with no driving external input and steady-state synaptic currents and 132 membrane potentials. The balance value is calculated by subtracting the sum of the 133 equilibrium synaptic inhibitory currents to the excitatory populations from the sum of 134 the equilibrium excitatory currents to the excitatory populations [4,38]. For example, 135 the balance value, B, for layer 2/3 is defined as follows: wherev E andv I are the equilibrium membrane potentials of the excitatory and inhibitory 137 populations, respectively.

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Fixed-point analysis 139 We evaluate the stability of the model by analysing the effects of small perturbations 140 of the parameters. The first step is to find equilibria of the network by setting the left 141 hand sides of the differential equations (i.e., all of the differential equations in the form 142 of Eq. (2) and Eq. (4) equal to zero and solving the following equations: To measure the paradoxical effect, the membrane potential of the inhibitory popula-160 tions is compared before and after applying the perturbation, where v I before is the sustained (non-transient) membrane potential of the inhibitory pop-162 ulation before perturbation and v I after is the sustained membrane potential of inhibitory 163 population after perturbation. A negative value of ∆v indicates a paradoxical response; 164 i.e., the membrane potential reduces after the external input is increased. We evaluate 165 changes in ∆v with variations in the parameters of the model.

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Excitation-inhibition balance 168 The balance in each motif between excitation and inhibition is evaluated separately for 169 each layer of the model.  Table 1.

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The balance values at the starred points are very close to zero, showing near equal 175 balance between excitation and inhibition in the motifs when the weights are set to the 176 neurophysiologically derived values given in Table 2 the neurophysiological findings of [22], which showed that the connectivity in layer 5 181 of the somatosensory cortex of a juvenile rat is excitatory dominated compared to the 182 superficial layers, such as layer 2/3.

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Fixed-point analysis of a single motif 184 We evaluate the stability of a motif when changing the connectivity weights.   Fixed-point analysis of the whole column 207 We also analyse the changes in the stability and firing rate of the neurons for different 208 values of connectivity weights across the whole column model. Fig. 4 shows the results 209 of this analysis when changing the connectivity weights of the inter-layer connections. 210 All other connections have the default values indicated in Table 1.   The layer 2/3 motif does not operate in the ISN regime: the membrane potential of 228 inhibitory neural population increases to a higher steady-state value after the perturbation 229 is applied. This is seen in Fig. 5A, which shows the membrane potentials of the inhibitory 230  of the paradoxical response to perturbation (negative magnitude of ∆v) (see Fig. 7A, B, 274 C) until a saddle node bifurcation occurs similar to the behaviors seen in Fig. 6. in Fig. 9 shows the bifurcation analyses when changing the inhibitory-to-inhibitory (II) 287 connectivity weights in isolated motifs.  The effects of varying the fixed parameters of the model were investigated to examine the 300 robustness of the model to these parameters. These included the connectivity constant, 301 k, and the saturation firing rates of the neural populations, φ E max and φ E max (see Table 2). 302 The default value of the external input was chosen as it generates firing rates close to 1 318 Hz, the observed spontaneous firing rates of neurons in cortex.

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To evaluate the effects of the parameters of the sigmoid function, we conducted 320 the same experiment using higher maximum firing rates for the neurons. The different 321 sigmoid function are shown in the panel inside Fig. 12A. The maximum firing rates were 322 increased to 50 Hz and 100 Hz; respective thresholds were also adjusted to 8 mV and 323 10.5 mV in order to maintain the same input-output relationship below 10 Hz firing 324 rate. The results in Fig. 12 show that the paradoxical behaviours of layer 2/3 (D and G) 325 and layer 4 (E and H) were largely unaffected by the changes to the sigmoid function as 326 the curves are similar to those in Fig. 12A and B, respectively. Layer 5 shows markedly 327 different behaviour with higher maximum firing rate, with the excitatory neurons in 328 layer 5 saturating at lower levels of external input, including at or close to the default 329

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We have also investigated the ranges for different connectivity weights in which the 359 network operates in the ISN regime and show the paradoxical response to perturbation 360 (Fig. 6, Fig. 8, Fig. 7 and Fig. 9) layer 5. This also accords to neurophysiological and anatomical studies [22].

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The fixed-point analysis 380 The results of the fixed-point analyses show that increasing the excitatory-to-excitatory 381 connectivity weight expands the regions where the fixed-point is bistable (Fig. 3A). 382 However, an increase in the value of the inhibitory-to-inhibitory connectivity weight 383 expands the region of having a stable fixed point in the network (Fig. 3A). On the other 384 hand, the data in Table 1 shows that the lowest values for the connectivity weights are the 385 recurrent excitatory-to-excitatory connections in each motif of a cortical column and the 386 highest value for connectivity weights are recurrent inhibitory to inhibitory connections. 387 This data also shows that in most cases the excitatory-to-inhibitory connections have 388 higher connectivity weights than inhibitory-to-excitatory connections, which accords with 389 the results obtained by the model that demonstrate a similar trend in the connectivity 390 weights in order to be in a stable regime.  The membrane potentials of the excitatory and inhibitory neural population in 428 layer 2/3: Synaptic currents from the inhibitory population in layer 2/3 to other populations: 431 The membrane potentials of excitatory and inhibitory populations in layer 4: Synaptic currents from the excitatory population in layer 4 to other populations: Synaptic currents from the inhibitory population in layer 4 to other populations: Membrane potentials of excitatory and inhibitory populations in layer 5: following equations for calculating the fixed points of the network: