The Standard Model of Type I Membranes
2.7 Dynamics of Inhibitory Networks
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Figure 2.6 – Identical firing characteristics in inhibitory networks (left) and excitatory-inhibitory networks (middle) in the balanced state. (A),(C) Spike patterns of 40 randomly chosen neurons, (B),(D) voltage traces of one random neuron, (E) firing rate distributions, (F) coefficient of variation distributions and (G) stationary voltage distributions for both types of networks, (parameters: N=10 000, K =100, ν¯ =1 Hz,J0=1,τm=10 ms,η=0.9,ε=0.3,NE =4NI).
networks were broad. The distributions of firing ratesνi and coefficients of variation cvi indicate substantial heterogeneities in the balanced networks. Moreover, these distributions were identical in both types of networks, due to the specific choice of coupling strengths, Eq. (2.31), that lead to identical input current statistics in the balanced state. This allows for a quantitative comparison of the dynamics of the two types of networks.
Although the balanced state emerged for a broad parameter range, two different transitions from the asynchronous to synchronous states were also observed in the theta neuron networks. In in-hibitory networks, the neurons’ activity synchronized upon increasing the connectivity K. Even though the spike sequences of individual neurons remained irregular, the population activity re-vealed a partially synchronous state forK>Kc≈200 (Fig. 2.7). In excitatory-inhibitory networks, the activity of neurons synchronized and became regular upon increasing the excitatory-inhibitory feedback loop strength ε above a critical valueεc (Fig. 2.8). The latter depended on the ratio of the excitatory coupling η = JJEE
IE between the excitatory and inhibitory neurons (Fig. 2.9). The excitatory interpopulation couplingJIE promoted this transition, whereas the excitatory intrapopu-lation couplingJEE hindered it. Since we here focus on the analysis of the asynchronous irregular balanced state, a high ratio η =0.9 is used throughout the rest of the paper. For this value, the transition to the synchronous state occurred atεc≈0.5.
2.7 Dynamics of Inhibitory Networks
Sparse inhibitory networks of theta neurons in the balanced state exhibit conventional determinis-tic chaos that is furthermore extensive (Fig. 2.10). We first examined the collective dynamics of sparse networks with fixed number of synapses per neuronKand increasing network size (number of neurons) N. The spike statistics was independent of N. The decreasing synchrony measure χ ∼1/√
N and the high coefficient of variation cv ≈0.8 demonstrate the typical asynchronous and irregular firing activity in these balanced networks (Fig. 2.10A,B). Importantly, the dynamics
2.7 Dynamics of Inhibitory Networks
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Figure 2.7–Transition from asynchronous irregular state (left) to synchronous irregular state (right) in inhibitory networks.(A) Spike patterns of 1000 randomly chosen neurons forK=100, (B) voltage trace of one random neuron forK=100, (C),(D) as (A),(B) but for forK=500 (other parameters: N=10 000, ν¯ =1 Hz,J0=1,τm=10 ms).
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Figure 2.8 –Transition from asynchronous irregular state (left) to synchronous regular state (right) in excitatory-inhibitory networks. (A) Spike patterns of 1000 randomly chosen neurons forε =0.3, (B) voltage traces of one random neuron for ε =0.3 (C),(D) as (A),(B) but for ε =0.6 (other parameters:
NE =8000,NI =2000,K=100,J0=1,τm=10 ms,η =0.9, the external currents were set according to the balance equation (2.25) with a target average firing rate ¯νbal=1 Hz).
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Figure 2.9–Phase diagrams of excitatory-inhibitory networks for three different ratiosη=JEE/JIE. (see coupling matrix (2.31)) Columns from left to right:η=1,0.9,0, from top to bottom: average firing rate ν, coefficient of variation cv and synchrony measure¯ χ; on the x-axis is plotted the target average firing rate according to the balance equation (2.25) and on the y-axis the excitatory-inhibitory feedback loop activation ε, (parameters:NE=8000,NI=2000,K=100,J0=1,τm=10 ms; displayed are averages of 10 runs with different network realizations).
is conventionally chaotic, indicated by positive and finite largest Lyapunov exponents (Fig. 2.10C).
Interestingly, the Lyapunov spectra were invariant to the network size. Plotting the Lyapunov expo-nents{λi}versus their indicesirescaled with the network sizeN, they converged to a unique shape for largeN (Fig. 2.10C). Consequently, the number of positive Lyapunov exponents, the entropy production rate H, Eq. (1.5), and the attractor dimension D, Eq. (1.6), increased linearly with N (Fig. 2.10D,E). This indicates extensive chaos and it is well justified to define the relative attractor dimensiond=D/N and the average entropy production rate per neuronh=H/N (Fig. 2.10F,G).
Both quantities were surprisingly high, symbolizing a high dimensional chaotic attractor in these networks and a rapid entropy production. To show that extensive chaos is a robust phenomenon in the studied networks, additional network size invariant Lyapunov spectra for different average firing rates ¯ν are presented in Fig. 2.10H-K.
Increasing the network-averaged firing rate ¯ν intensified the chaos in the balanced networks (Fig. 2.11). Although the irregularity in the neurons’ spiking activity decreased, the largest Lya-punov exponent, the number of positive LyaLya-punov exponents, attractor dimension and entropy pro-duction rate increased with increasing firing rate. We plotted the entropy propro-duction rate divided by the average firing rate in Fig. 2.11G, providing an estimate of the average entropy production per spike per neuron. The surprisingly high rate of entropy production of up to 1 bit per spike per neuron should be contrasted with the actual information content in cortical neurons supplied by
2.7 Dynamics of Inhibitory Networks
Figure 2.10 – Extensive chaos in balanced inhibitory networks of N theta neurons. (A) Synchrony measure χ (straight line: χ∼1/√
N), (B) average coefficient of variation cv, (C) full Lyapunov spectra {λi}, (D) attractor dimension D, Eq. (1.6), (straight line: D=0.18N), (E) entropy production rate H, Eq. (1.5), (straight line: H=0.8N), (F) attractor dimension in percent of phase space dimensiond=D/N and (G) entropy production rate per neuronh=H/N, (H)-(K) full Lyapunov spectra for different firing rates ν, (τ¯ m=10 ms; displayed are averages of 10 runs with different network realizations).
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Figure 2.11 – Deterministic chaos in balanced inhibitory networks for varied average firing rates ν.¯ (A) Synchrony measureχ, (B) average coefficient of variation cv, (C) full Lyapunov spectra {λi}, (D) largest Lyapunov exponentλmax=λ1, (E) mean Lyapunov exponentλmean=N1∑iλi, (F) attractor dimension in percent of phase space dimensiond=D/Nand (G) entropy production rate per neuronh=H/N, (τm= 10 ms; displayed are averages of 10 runs with different network realizations).
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Figure 2.12–Deterministic chaos in balanced inhibitory networks for varied coupling strengths J0. (see Fig. 2.11 for description)
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Figure 2.13–Deterministic chaos in balanced inhibitory networks for varied number of presynaptic neurons K. A transition from the asynchronous irregular balance state to a synchronous irregular state occurs at a network size independentKc≈200 (see Fig. 2.11 for description).
sensory input streams. This was experimentally estimated to be of order 1 bit per spike per neuron as well [93, 94]. Our results thus imply a rapid loss of input information in such chaotic networks.
Does the result of a rapid information loss strongly depend on other parameters in our model?
The membrane time constantτm is set to 10 ms throughout this paper, which is the physiological relevant time scale for membrane time constants. Since it is ’just’ the unit of any time-related quantity in our model, changing τm would just change the time-related quantities accordingly.
For example, doubling τm to 20 ms, would result in half the average firing rates and Lyapunov exponents in Fig. 2.11. The attractor dimension and the average entropy production rate per spike, however, would not be affected by a change of the membrane time constant.
The influence of the coupling strengthJ0 is displayed in Fig. 2.12. A vanishing J0 would cor-respond to the uncoupled case, in which these fully deterministic systems would exhibit stable regular dynamics. IncreasingJ0 led to the asynchronous, irregular, chaotic balanced state. While the quantitative values of the Lyapunov exponents depended on the coupling strength, the syn-chrony measure, coefficient of variation, attractor dimension and entropy production rate appeared to saturate atJ0of the order of one.
Surprisingly, a large number of synapses per neuronKled to a transition from the asynchronous chaotic state to a synchronous chaotic state in inhibitory networks (Fig. 2.13). This behavior was rather unexpected, as the statistics of the balanced state should be independent ofK in the large