Small changes in synaptic gain lead to seizure-like activity in neuronal network at criticality

Bioinformatics

Architecture of the LCM

For the reader’s convenience, we introduce the structure of the LCM (see Fig. 1). The LCM represents a small region in the visual cortex of the cat as a sheet of cortical columns, within which neuron groups are situated (see Methods and Fig. 1B). Neuronal dynamics are modelled using a ‘mean field approximation’ approach, which treats the same type of neurons within a column as a group acting as a single entity within the network. The dynamics and connections of single neurons in a neuron group are averaged using the mean field approximation14,16. We expanded the original LCM to incorporate a thalamocortical loop. Hence, the model now incorporates eleven neuron types in the cortex and three neuron types in the thalamus (see Fig. 1C,D and Table S1). Interactions between neuron groups are controlled by synaptic connections within the cortex and the thalamus (see Fig. 1C and Table S5). The LCM models the stimulus input to the thalamus as point spike sources, with spike rates generated using a Monte Carlo method (see Methods). For each run of the LCM, the membrane potentials of all neuron groups are first initialised at their resting values (−65 V), and then updated every 0.5 msec through five processes: action potential (AP) generation, AP conduction and distribution, synaptic transmission, postsynaptic potential (PSP) aggregation, and membrane potential (MP) formation (see Fig. 1D). Each of these processes is modelled using a set of equations elaborated upon in the Methods. The results reported below were generated by an instantiation of the LCM configured to simulate a cortical area of 1.12 × 1.12 mm2, which contains 20 × 20 cortical columns with the size of 56 μm17. The results were produced from a 5.12-second recording of neuronal firing rates after 10 seconds of running of the model.

Figure 1

Architecture of the LCM. Illustrated are (A) the visual pathway in cats, (B) the structure of the LCM, (C) the synaptic connections between neurons simulated in the LCM, and (D) the neuronal processes simulated in the LCM. In Figure (C), the arrows indicate the direction of synaptic transmission; and sizes of squares and circles indicate the relative densities of corresponding neuron groups (not to scale). Only a partial synaptic connection map is shown; a complete map is provided in Table S5 in the Supplementary Information. Acronyms: SI–sensory input; TH–thalamus; AP–action potential; PSP–postsynaptic potential; MP–membrane potential; E1–excitatory neurons in layer I of the cortex; I1–interneurons in layer I; P2/3–pyramidal neurons in layer II and III; I2/3–interneurons in layer II and III; P4–pyramidal neurons in layer IV; SS4–spiny stellate neurons in layer IV; I4–interneurons in layer IV; P5–pyramidal neurons in layer V; I5–interneurons in layer V; P6–pyramidal neurons in layer VI, I6–interneurons in layer VI; IRTN–interneurons in RTN of the thalamus; RLGN–relay neurons in LGN of the thalamus; ILGN–interneurons in LGN.

Temporal features of different states of neuronal activity

Parameters used to model the behaviour of neuron groups were derived from published experimental results (See Tables S1S5). We found that the excitatory and inhibitory synaptic gains (gE and gI), which respectively control the efficiency of excitatory and inhibitory synaptic transmission, dramatically influence neuronal firing rates. Mathematically, synaptic gain can be expressed as (also see Equation (7)),

$${g}_{p}=frac{PS{P}_{qp}}{{varphi }_{p}{N}_{qp}}f({V}_{q},{varphi }_{p})$$

(1)

where PSPqp is the amplitude of PSPs at an afferent spike rate ϕp, Nqp is the number of synapses, and f(Vq, ϕp) is a scaling factor for spike adaptation and membrane potential dependency (refer to Equation (7) for more details). From a network perspective, excitatory and inhibitory synaptic gain respectively reflect the excitatory and inhibitory connection strength between neurons in the network with changes in gain corresponding to the effects of short-term synaptic plasticity.

Figure 2 displays the neuronal firing rates for a range of synaptic gains. Three types of neuronal activity with markedly different firing rates and oscillation states were noted. Low amplitude asynchronous activity occurred at low excitatory synaptic gain [<0.38 uV/(mV · Hz)] or high inhibitory synaptic gain (see Fig. 2). Here, the mean firing rate was less than one AP/sec. Neuronal activity was of low frequency with no specific frequency being dominant. High amplitude asynchronous activity was produced when the excitatory synaptic gain was high and the inhibitory gain was small. The mean firing rate exceeded 60 AP/sec, and in the frequency spectrum of neuronal firing rates, low-frequency oscillations were enhanced. When both excitatory and inhibitory synaptic gain were large, strong rhythmic activity occurred. In this state, the neurons were highly activated (mean firing rates >10 AP/sec), and the frequency spectra showed obvious peaks.

Figure 2
Figure 2

Neuronal firing rates with different synaptic gains. The plots in panel (A) displays firing rates simulated without (black lines) and with stimulus (red lines) using a range of excitatory and inhibitory synaptic gains. The excitatory and inhibitory synaptic gain values are shown on the top and the left, respectively [unit: uV/(mV · Hz)]. The number on the right of each figure indicates the value at the middle of the vertical scale bars (unit: AP/sec). The vertical scale bars on the right represent a firing rate of 20 AP/sec, and the horizontal scale bars at the bottom represent a period of 500 msec. Six amplified firing rate traces are plotted below the panel. The figures in panels (B and C) show the power spectrum densities (PSD) and magnitude-squared coherence (MSC) of six typical neuronal firing rates in panel (A). Corresponding plots are indicated by the lower case letters (a–c etc.). See also Fig. 3.

Figure 3 displays four quantities calculated from the neuronal firing rates obtained using different synaptic gains: temporal mean firing rates (MFR), the mean power spectrum densities (PSD) of the low (2–15 Hz, PSDL) and the high (16–50 Hz, PSDH) frequency bands, the mean magnitude-squared coherence (MSC) of the low (MSCL) and the high (MSCH) frequency band. The first three measurements quantify temporal properties of neuronal firing and the MSCs measure coherence between neuron firing in columns. In Fig. 4, parameters are plotted against excitatory and inhibitory synaptic gains. Figures 3 and 4 reveal a complex relationship between neuronal firing rate and synaptic gains. The firing rate generally increases as excitatory synaptic gain (gE) increases or inhibitory synaptic gain (gI) decreases in magnitude. However, the change in firing rate is not continuous; a series of critical synaptic gain combinations are observed, around which small changes in excitatory or inhibitory gain cause dramatic jumps in the firing rate. For example, with an inhibitory gain of 1 uV/(mV · Hz), the firing rate jumps from about 0.3 AP/sec to >60 AP/sec as excitatory gain increases from 0.394 to 0.397 uV/(mV · Hz) (refer to Fig. 3D). On the synaptic gain maps shown in Fig. 3, the critical combinations of excitatory and inhibitory synaptic gains fall along a line:

$${g}_{{rm{E}},{rm{C}}}=0.018{g}_{{rm{I}}}+0.376,,{rm{or}},{g}_{{rm{I}},{rm{C}}}=55.6{g}_{{rm{E}}}+20.89,$$

(2)

where gE,C and gI,C are the critical excitatory and inhibitory gains, respectively. Secondly, this line divides neuronal firing rates into two regions: Region 1: where gE < gE,C and gI > gI,C, in which the neurons fire at low rate (mean firing rate <0.4 AP/sec), and Region 2: where gE > gE,C and gI < gI,C, in which the neurons fire at medium to high rates (mean firing rate >10 AP/sec, refer to Fig. 4A,B). Neuronal firing rates in the two regions exhibit different dependencies on synaptic gains. In Region 1, the firing rate is positively correlated with both the excitatory (Pearson’s correlation coefficient r = 0.65) and the inhibitory gain (r = 0.38). In Region 2, firing rate is weakly negatively correlated with excitatory gain (r = −0.31), but strongly negatively correlated with inhibitory gain (r = −0.93). Neuronal firing rate increases exponentially as the magnitude of inhibitory gain decreases. The firing rate increases from about 20 AP/sec at gI = 6 uV/(mV · Hz) to 40 AP/sec at gI = 2 uV/(mV · Hz) and about 100 AP/sec at gI = 0.2 uV/(mV · Hz). Excitatory gain does not affect the firing rate significantly (see Fig. 4B), but the range of firing rates increases slightly with a decrease in the lower limit (refer to Fig. 4A).

Figure 3
Figure 3

Neuronal firing rate properties with different synaptic gains. The plots display the mean firing rates (MFR) without stimulation (A), with stimulation (B), their differences (C) and dependency on excitatory gain for three typical inhibitory gains (D); the mean power spectrum density over firing rates of 2–15 Hz (PSDL) without stimulation (E), with stimulation (F), their differences (G) and dependency on excitatory gain (H); the mean power spectrum density over firing rates of 16–50 Hz (PSDH) without stimulation (I), with stimulation (J), their differences (K) and dependency on excitatory gain (L); and the mean magnitude-squared coherence for firing rates of 2–15 Hz (MSCL) without stimulation (M), with stimulation (N), their differences (O) and dependency on excitatory gain (P); the mean magnitude-squared coherence for firing rates of 16–50 Hz (MSCH) without stimulation (Q), with stimulation (R), their differences (S) and dependency on excitatory gain (T). Discrete values for synaptic gain were used with minimum intervals of 0.003 uV/(mV · Hz) and 0.1 uV/(mV · Hz) between adjacent values for excitatory and inhibitory synaptic gain respectively. On the right column, the lines with open and closed markers display results for datasets without and with stimulus, respectively. Error bars on the line plots show the standard deviations (SD) for 10 runs of the LCM with the same parameter values but different random number kernels. See also Fig. 4.

Figure 4
Figure 4

Neuronal firing rate measurements plotted versus synaptic gains. Shown are the mean firing rates (MFR; (A,B)), the mean power spectrum density over 2–15 Hz (PSDL; (C and D)), the mean power spectrum density over 16–50 Hz (PSDH; (E and F)), the mean magnitude-squared coherence over 2–15 Hz (MSCL; (G,H)), the mean magnitude-squared coherence over 16–50 Hz (MSCH; (I and J)) of neuronal firing rates plotted versus excitatory gain (A,C,E,G and I) and inhibitory gain (B, D,F,H and J) for conditions when excitatory gains were smaller than the critical values (gE ≤ gE,C − δ, where δ = 0.002 uV/(mV · Hz)) and when excitatory gains were larger than the critical values (gE ≥ gE,C + δ). Each dot on the plot represents a result obtained using one synaptic gain pair. The inhibitory synaptic gain values in figure (A,C,E,G and I) and the excitatory synaptic gain values in figure (B,D,F,H and J) are represented by colour, and the corresponding colour bars are shown on the right. The Pearson’s correlation coefficient (r) was calculated for each case. The data was produced from non-stimulated datasets, as shown in Fig. 3.

In Region 1 (gE < gE,C), oscillations in both low and high frequency bands increased with excitatory gain (r = 0.78 for low frequency band; and r = 0.87 for high frequency band; refer to Fig. 4C and E), and were not significantly affected by changes in inhibitory gain (r = −0.1 for low frequency band; and r = 0.11 for high frequency band; refer to Fig. 4D and F). In Region 2 (gE > gE,C), the PSDs remained low [<0.1 (AP/sec)2/Hz] for excitatory gain <0.43 uV/(mV · Hz). When excitatory gain exceeded 0.43 uV/(mV · Hz), the PSDs varied between 0–2 (AP/sec)2/Hz for the low-frequency band and 0–8 (AP/sec)2/Hz for the high-frequency band. As with neuronal firing rate, the PSDs were more strongly dependent on inhibitory gain than on excitatory gain (refer to Fig. 4C–F). PSDs in both low and high-frequency bands increased as inhibitory gain increased, implying that strong neuronal oscillations require high inhibitory activity. The PSDs of high but not low-frequency bands appeared to saturate at gI = 3 uV/(mV · Hz) (refer to Fig. 4F).

The MSCs in both low and high-frequency bands did not vary significantly with either excitatory or inhibitory gains in Region 1 (gE < gE,C). In Region 2 (gE > gE,C), the correlations between the MSCs and excitatory gains were moderate in the low-frequency band (r = 0.59) and low in the high-frequency band (r = 0.27). The MSCs had a strong, non-linear relationship with the inhibitory gains in the region. Highly coherent neuronal firing occurred at high excitatory gain [gE ≥ 0.43 uV/(mV · Hz)] and moderately low inhibitory gain (between 1.3 and 3.3 uV/(mV · Hz); refer to Fig. 4G,H).

The LCM incorporates spike inputs from two external sources: cortico-cortical projections to cortical neurons and sensory inputs to thalamic neurons. Cortico-cortical input was modelled as low-magnitude (1 AP/sec), low-frequency (frequency cut off at 20 Hz) white noise resembling temporal features of alpha activity generated in the absence of a stimulus. Afferent thalamic input was modelled for two conditions: (1) the non-stimulated state, modelled as low magnitude (1 AP/sec) low frequency (frequency cut off at 20 Hz) white noise; and (2) the stimulated state, modelled as high magnitude (50 AP/sec) high frequency (frequency cut off at 50 Hz) white noise. Cortico-cortical connections to each column were treated as being independent (i.e., asynchronous input) whereas thalamic inputs were the same for all columns (i.e., synchronous input). The neuronal firing rates and their response to stimulation are shown in Figs 2 and 3. Stimulation did not change mean firing rates significantly when ({g}_{{rm{E}}}gg {g}_{{rm{E}},{rm{C}}}) or ({g}_{{rm{E}}}ll {g}_{{rm{E}},{rm{C}}}) but firing rate increased by as much as 80 AP/sec when gE was close to gE,C (see Fig. 3C). Stimulation did not affect neuronal oscillation or synchronisation in Region 1 (gE < gE,C). In Region 2 (gE > gE,C), stimulation changed PSDs of neuronal firing in both low and high-frequency bands only when gI > 2.4 uV/(mV · Hz) (Fig. 3G and K) and it significantly increased MSCs of neuronal firing in both low and high-frequency bands when gI < 0.4 uV/(mV · Hz) (refer to Fig. 3O and S).

Response to transient changes in synaptic gain

The results shown above were simulated using fixed synaptic gains. However, the efficiency of synaptic transmission may be influenced by many factors. We therefore examined the changes in neuronal firing rates induced by slight, transient changes in synaptic gain (see Fig. 5). For most combinations of synaptic gains, a change in firing rate occurred within milliseconds and neuronal activity returned to its previous state as soon as gains were restored to their initial values. The transition was more complex, however, for certain combinations of synaptic gain. For example, for the combination of gE = 0.4 uV/(mV · Hz) and gI = 2 uV/(mV · Hz) (see Fig. 5), the effects of changes in excitatory gain persisted for hundreds of milliseconds. For the combination of gE = 0.403 uV/(mV · Hz) and gI = 2 uV/(mV · Hz), a 0.015 uV/(mV · Hz) increase in excitatory gain resulted in significant changes in neuronal activity only after a 1-second delay with a strong neuronal oscillation that lasted almost 0.5 second before steady state was reached. Neuronal activity did not return to its original state after synaptic gains were restored to their initial values. Similar behaviours were also observed with temporary changes in inhibitory synaptic gain. It should be noted that the complex transitions in neuronal activity occurred when gE increased from below gE,C to above gE,C or gI decreased from above gI,C to below gI,C.

Figure 5
Figure 5

Neuronal firing rates under a small transient change in synaptic gains. (A) The red lines display the firing rates when the excitatory synaptic gains, which are shown on the left, are increased by 0.015 uV/(mV · Hz) for 2 seconds (indicated by solid vertical lines). The black lines display neuronal firing rates without the change in synaptic gain. Inhibitory gains are fixed at 2 uV/(mV · Hz) and the corresponding critical excitatory gain is ~0.431 uV/(mV · Hz). (B) The red lines display the firing rates when inhibitory synaptic gains, which are shown on the left, are reduced by 0.2 uV/(mV · Hz) (i.e., weaker inhibition) for 2 seconds (indicated by solid vertical lines). The black lines display neuronal firing rates without the change in synaptic gain. The excitatory gains are fixed at 0.431 uV/(mV · Hz) and the corresponding critical inhibitory gain is about 3 uV/(mV · Hz). The number on the right is the firing rate at the middle of the vertical scale bar (unit: AP/sec), and markers represent the values of vertical scale bars: •0.1 AP/sec, ••0.5 AP/sec, +−5 AP/sec; +25 AP/sec, ++−50 AP/sec. For comparison, the same random number kernels were used for all cases.

Response to changes in the reversal potential of inhibitory synapses

We also examined neuronal network response to changes in the reversal potentials of inhibitory synapses. Figure 6 displays neuronal firing rates obtained using different combinations of inhibitory synaptic gains and reversal potentials of inhibitory synapses. We observed a boundary in the map, around which a small positive shift in the reversal potential led to a jump in neuronal firing rates (Fig. 6A). The critical values of the reversal potentials moved positively when inhibitory gain increased. In addition, neuronal oscillations did not change significantly with the reversal potential under weak inhibition but the oscillation amplitudes in both low- and high-frequency bands increased significantly when reversal potentials shifted positively (Fig. 6B,C).

Figure 6
Figure 6

Neuronal firing rate measurements under a small shift in the reversal potentials of inhibitory synapses. Shown are the mean firing rates (MFR; (A)), the mean power spectrum density over 2–15 Hz (PSDL; (B)), and the mean power spectrum density over 16–50 Hz (PSDH; (C)) simulated using combinations of inhibitory synaptic gains (gI) and reversal potentials of inhibitory synapses (({{V}_{I}}^{[{rm{rev}}]})).

Neuronal activity with changes associated with SCN1A mutations

The LCM can be used to analyse the behaviour of the network with changes in the biophysical properties or synaptic connections of neurons. We examined the effects of biophysical changes associated with SCN1A gene mutations. This gene encodes the alpha 1 subunit (Nav1.1) of the neuronal voltage-gated sodium channel. Mutations are associated with a wide range of epilepsy syndromes, ranging from relatively mild simple febrile seizure (FS) and Generalised Epilepsy with Febrile Seizures Plus (GEFS+) to more severe Intractable Childhood Epilepsy with Generalized Tonic-Clonic seizures (ICEGTC) and Dravet syndrome (DS)18,19,20,21,22. A variety of biophysical changes have been associated with different mutations, including impaired firing capability and positive shifts in the firing thresholds of inhibitory neurons23,24,25.

Impaired firing capability was simulated by gradually reducing maximum firing rates ({{F}_{{rm{I}}}}^{[{rm{max }}]}) (see Equation (5)) from 200 to 40 AP/sec, in steps of 10 AP/sec, in all inhibitory neuron groups. Figure 7 displays the neuronal firing rates with reduced ({{F}_{{rm{I}}}}^{[{rm{max }}]}) and different synaptic gains. As expected, the neuronal firing rate increased as ({{F}_{{rm{I}}}}^{[{rm{max }}]}) decreased for all synaptic gains tested, but the firing rate did not change continuously. It jumped from a low rate to a high rate around certain values of ({{F}_{{rm{I}}}}^{[{rm{max }}]}), and the closer gE was to gE,C, the smaller the reduction in ({{F}_{{rm{I}}}}^{[{rm{max }}]}) required for an abrupt transition (refer to Fig. 7E). PSDs at both high and low frequency increased as ({{F}_{{rm{I}}}}^{[{rm{max }}]}) was reduced and there was an abrupt transition at the same ({{F}_{{rm{I}}}}^{[{rm{max }}]}) as that at which the abrupt transition in firing rates was observed. PSDs at both high and low frequency decreased when ({{F}_{{rm{I}}}}^{[{rm{max }}]}) was extremely low (({{F}_{{rm{I}}}}^{[{rm{max }}]}) < 70 AP/sec).

Figure 7
Figure 7

Neuronal firing rates with reduced firing capability of inhibitory neuronal groups. Shown are the neuronal firing rates when the maximum firing rate of inhibitory neuron groups (({{F}_{{rm{I}}}}^{[{rm{max }}]})) is reduced. Figures (AD) show the firing rates simulated using four combinations of synaptic gains. The synaptic gain values are shown on the left [unit: uV/(mV · Hz)] and ({{F}_{{rm{I}}}}^{[{rm{max }}]}) values are shown on the top of the figures (unit: AP/sec). The number on the right is the firing rate at the middle of the vertical scale bar (unit: AP/sec), and the markers indicate the values of vertical scale bars: •0.1 AP/sec; +−5 AP/sec; +25 AP/sec. The horizontal scale bars represent a 200 millisecond period. Figures (EG) display the temporal mean firing rates (MFR; (E)), and the mean power spectrum density for firing rates of 2–15 Hz (PSDL; (F)) and 16–50 Hz (PSDH; (G)). The synaptic gain values are displayed on the top [unit: uV/(mV · Hz)]. The error bars represent the standard deviation for 10 runs of the LCM with the same parameter values but different random number kernels.

A positive shift in the firing thresholds of inhibitory neurons was simulated by gradually increasing the voltage at half-maximum-firing (({{V}_{{rm{I}}}}^{[{rm{HMF}}]}); see Equation (5)) for all inhibitory neuron groups from −45 mV to −19 mV in steps of 2 mV. Figure 8 displays the neuronal firing rates with changes in ({{V}_{{rm{I}}}}^{[{rm{HMF}}]}) and different synaptic gain combinations. Positive shifts in firing thresholds caused the neuronal firing rate to increase discontinuously. Even a small, 2–4 mV, shift in neuronal firing threshold caused a significant increase in the firing threshold (refer to Fig. 8E). The PSDs at both high and low frequencies increased after a small positive shift in firing threshold, but they returned to minimum values for ({{V}_{{rm{I}}}}^{[{rm{HMF}}]}) > −29 mV.

Figure 8
Figure 8

Neuronal firing rates simulated using positively shifted firing thresholds of inhibitory neuron groups. Shown are the neuronal firing rates simulated using positively shifted voltage at half-maximum firing (({V}_{{rm{I}}}^{[{rm{HMF}}]})). Figures (AD) display the firing rates obtained using four synaptic gain combinations. The synaptic gain values are shown on the left [unit: uV/(mV · Hz)], and the values for ({V}_{{rm{I}}}^{[{rm{HMF}}]}) are shown on the top (unit: mV). The number on the right is the firing rate in the middle of the vertical scale bar (unit: AP/sec), and the markers indicate the values of vertical scale bars: •0.1 AP/sec; ••0.5 AP/sec; +−5 AP/sec; +25 AP/sec. The horizontal scale bars represent a 200 millisecond period. Figures (EG) display the temporal mean firing rates (MFR; (E)), and power spectrum density for firing rates of 2–15 Hz (PSDL; (F)) and 16–50 Hz (PSDH; (G)) using different synaptic gains. Synaptic gain values are displayed at the top [unit: uV/(mV · Hz)]. Error bars represent the standard deviation for 10 runs of the LCM using the same parameter values but different random number kernels.

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