Dynamics of Excitatory-Inhibitory Networks

Figure 2.14 – Chaotic dynamics in balanced excitatory-inhibitory networks while activating the excitatory-inhibitory feedback loops withε. (values of inhibitory networks as dotted lines for compari-son) (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, (pa-rameters:NI=2000,NE =8000,K=100, ¯νE =ν¯I=1 Hz,J₀=1,τ_m=10 ms,η=0.9,ε=0.3; displayed are averages of 10 runs, with different network realizations, in (C) averages of 3 runs).

K-limit, due to the specific square root scaling of the external currents and coupling strengths.

However, increasingKwhile assuring the same connection probability between neurons p=K/N, destabilized the asynchronous state. The occurrence of this transition at a critical connectivityK_c≈ 200 that was independent ofNas indicated by the results shown in Fig. 2.13A for p=0.05 (filled circles) and p=0.01 (open circles). The asynchrony forK<200 was confirmed by a decreasing synchrony measureχ for increasingN. Such a transition, with a network size independent critical connectivity was previously reported in networks of inhibitory hippocampal interneurons [38].

One should note that the irregularity of the individual spike trains was insensitive to the transition.

The asynchronous and the synchronous state displayed identical, high coefficients of variation.

The collective dynamics of the inhibitory networks were only slightly influenced by the transition from asynchrony to synchrony as well. In the asynchronous state, the largest Lyapunov exponent and the entropy production rate slowly increased, the attractor dimension slightly decreased with increasingK. In the synchronous state, the largest Lyapunov exponent continued to increase and the information dimension continued to decrease, whereas the entropy production rate changed to slowly decrease with increasingK.

2.8 Dynamics of Excitatory-Inhibitory Networks

Realistic cortical networks typically consist of 20% inhibitory and 80% excitatory neurons [95].

We therefore investigated the influence of excitatory coupling on the dynamics of the balanced state (Fig. 2.14). For a quantitative comparison with inhibitory networks, the coupling strengths were chosen as explained above, Eq. (2.31),

J= J₀

With this parametrization, the magnitude of input current fluctuations remained unchanged while increasing the strength of the excitatory-inhibitory feedback loops withε. Thus the firing statistics

0 0.3 0.6

Figure 2.15–Stable dynamics during transition from asynchronous irregular to synchronous regular state for strong excitatory couplingsε. (values of inhibitory networks as dotted lines for comparison) (A) Synchrony measure χ, (B) average coefficient of variation cv, (C) largest Lyapunov exponent, (D) mean Lyapunov exponent, (E) attractor dimension in percent of phase space dimension, (F) entropy production rate per neuron, (G) actual population-averaged firing rates, (parameters: N_I =2000,N_E=8000,K=100, J₀=1,τ_m=10 ms,η=0.9, the input currents were here chosen to fulfill the balance equation (2.25) for a certain target average firing rate ¯νbal; displayed are averages of 3 runs with different network realizations).

remained unchanged while increasing ε (Fig. 2.14A,B). For ε =0, all excitatory neurons were passive, in the sense that they did not provide any feedback to the network and their dynamics was dominated by the inputs from the inhibitory neurons. The Lyapunov spectrum was therefore very similar to an equivalent spectrum of a network of exclusively inhibitory neurons of one fifth size (dotted line in Fig. 2.14C). In fact, the positive part was identical, which suggests that the unstable modes were due to the active inhibitory neurons. Consequently, the largest Lyapunov exponent was identical to the value of the inhibitory networks (Fig. 2.14D), while the attractor dimension and entropy production rate atε =0 were one fifth of the inhibitory values (Fig. 2.14F,G). Upon acti-vation of the excitatory-inhibitory feedback loops withε>0, the chaos in the networks increased.

Although the largest Lyapunov exponent hardly increased in magnitude, the attractor dimension and entropy production showed a strong dependence on the excitatory coupling. The increasing attractor dimension seemed to saturate below the inhibitory value, while the entropy production increased linearly, but did not exceed the values of inhibitory networks (dotted lines), because the aforementioned transition to the synchronous state occurred beforehand. It is thus interesting to see, how this is affected by the transition to the synchronous state.

During the transition from the asynchronous irregular state to a synchronous regular state in excitatory-inhibitory networks the firing statistics and the collective dynamics changed consider-ably (Fig. 2.15). The sudden change to χ =1 and cv=0 indicate the synchronous and regular firing activity above ε_c, whose precise value depended on the average firing rate ¯ν. During the transition to the synchronous state, the actual population-averaged firing rates increased abruptly, and strongly deviated from the expected firing rates ¯ν_bal from the balance equation (2.25). The largest Lyapunov exponent and entropy production rate vanished, and the attractor dimension be-came one. This indicates stable periodic dynamics during this transition. Note, that the entropy production rate and attractor dimension never increased the values of inhibitory networks (dotted lines). We will now continue with the description of the dynamics of the asynchronous irregular balanced state.

Increasing the number of excitatory neurons in networks with a fixed number of inhibitory

neu-2.8 Dynamics of Excitatory-Inhibitory Networks

Figure 2.16 – Chaotic dynamics in balanced excitatory-inhibitory networks for varied number of excitatory neurons NE. (values of inhibitory networks as dotted lines for comparison) (A) Synchrony measure χ, (B) average coefficient of variation cv, (C) largest Lyapunov exponent, (D) mean Lyapunov exponent, (E) attractor dimension in percent of phase space dimension and (F) entropy production rate per neuron, (parameters:NI=2000,NE=8000,K=100, ¯νE=ν¯I=1 Hz,J₀=1,τ_m=10 ms,η=0.9,ε=0.3;

displayed are averages of 10 runs with different network realizations).

ronsN_I =2000 reduced the intensity of the chaos (Fig. 2.16). We increased the number of excita-tory neurons fromN_E=0 toN_E=10 000. The excitatory-inhibitory feedback loop strength was set toε=0.3. Again, the spike statistics did not change, while the dynamics changed in a way that the attractor dimension and entropy production decreased with increasing number of excitatory neu-rons. The largest Lyapunov exponent was rather unaffected. The high ratio of excitatory neurons in realistic cortical networks might thus be beneficial in terms of an reduced entropy production and hence a slower loss of input information.

The variation of the average firing rate revealed a qualitatively similar dependence of the dy-namics in excitatory-inhibitory networks as observed in inhibitory networks (Fig. 2.17). Varying the population-averaged firing rates ¯νE =ν¯I =ν¯ did not change the firing statistics in excitatory-inhibitory networks compared to the excitatory-inhibitory networks. This was expected from the construction of the coupling matrix that led to the same input current statistics in the balanced state in both types of networks. The largest and the mean Lyapunov exponent were also independent of the type of network. Nevertheless, the Lyapunov spectra changed slightly in excitatory-inhibitory networks, such that the attractor dimension and entropy production rate were slightly reduced as discussed above, but showed qualitatively the same dependence on the average firing rate.

Increasing the number of incoming connections per neuronK led to similar dynamics in both types of networks as well, although the transition to the synchronous irregular state occurred at much higher K_c in excitatory-inhibitory networks (Fig. 2.18). We checked two ratiosN_I =20K (filled circles) and N_E =20K (open squares), while 80% of the neurons were excitatory and 20% were inhibitory (N_E =4N_I) in both cases. The synchronous irregular state emerged for K>K_c≈1000, which is a five times higher critical connectivity compared to exclusively inhibitory networks. In the asynchronous irregular state, the firing statistics, Lyapunov exponents, attractor dimension and entropy production rate were merely independent of the number of synapses K.

This was expected since the statistics of the balanced state is independent ofK.

Investigating the dependence of the network dynamics on the number of neuronsN, we observed extensive chaos in excitatory-inhibitory networks (Fig. 2.19). The firing statistics, captured in the

0 5 10

Figure 2.17 – Chaotic dynamics in balanced excitatory-inhibitory networks for varied population-averaged firing ratesν¯E=ν¯I=ν.¯ (see Fig. 2.16 for description).

decreasing synchrony measureχ ∼1/√

N and constant high coefficients of variation cv, indicate the typical asynchronous irregular balanced state. The Lyapunov spectra converged to a system size independent shape. Hence the number of positive Lyapunov exponents, attractor dimension and entropy production rate scaled linearly with the number of neurons. This suggests a well-defined thermodynamic limit of the dynamics of excitatory-inhibitory networks in the balanced state and confirms the results from inhibitory networks. Altogether, one can conclude that inhibitory net-works of theta neurons already captured the dynamics of excitatory-inhibitory netnet-works very well.

Both types of networks exhibit extensive deterministic chaos in the balanced state.