Asynchronous and Coherent Dynamics in Balanced Excitatory-Inhibitory Spiking Networks

Abstract

Dynamic excitatory-inhibitory (E-I) balance is a paradigmatic mechanism invoked to explain the irregular low firing activity observed in the cortex. However, we will show that the E-I balance can be at the origin of other regimes observable in the brain. The analysis is performed by combining extensive simulations of sparse E-I networks composed of N spiking neurons with analytical investigations of low dimensional neural mass models. The bifurcation diagrams, derived for the neural mass model, allow us to classify the possible asynchronous and coherent behaviors emerging in balanced E-I networks with structural heterogeneity for any finite in-degree K. Analytic mean-field (MF) results show that both supra and sub-threshold balanced asynchronous regimes are observable in our system in the limit N >> K >> 1. Due to the heterogeneity, the asynchronous states are characterized at the microscopic level by the splitting of the neurons in to three groups: silent, fluctuation, and mean driven. These features are consistent with experimental observations reported for heterogeneous neural circuits. The coherent rhythms observed in our system can range from periodic and quasi-periodic collective oscillations (COs) to coherent chaos. These rhythms are characterized by regular or irregular temporal fluctuations joined to spatial coherence somehow similar to coherent fluctuations observed in the cortex over multiple spatial scales. The COs can emerge due to two different mechanisms. A first mechanism analogous to the pyramidal-interneuron gamma (PING), usually invoked for the emergence of γ-oscillations. The second mechanism is intimately related to the presence of current fluctuations, which sustain COs characterized by an essentially simultaneous bursting of the two populations. We observe period-doubling cascades involving the PING-like COs finally leading to the appearance of coherent chaos. Fluctuation driven COs are usually observable in our system as quasi-periodic collective motions characterized by two incommensurate frequencies. However, for sufficiently strong current fluctuations these collective rhythms can lock. This represents a novel mechanism of frequency locking in neural populations promoted by intrinsic fluctuations. COs are observable for any finite in-degree K, however, their existence in the limit N >> K >> 1 appears as uncertain.

Keywords: asynchronous dynamics; balanced spiking neural populations; coherent chaos; collective oscillations; neural mass model; quadratic integrate and fire neuron; sparse inhibitory-excitatory networks; structural heterogeneity.

Figure 1

Bifurcation diagrams of the neural mass model. The bifurcation diagrams concern the dynamical state exhibited by the excitatory population in the bidimensional parameter spaces $(I_0^{(e)},{\Delta }_0^{(ee)})$ (A), $(K,{\Delta }_0^{(ee)})$ (B), and $({\Delta }_0^{(ii)},{\Delta }_0^{(ee)})$ (C). The regions marked by Roman numbers correspond to the following collective solutions: (I) an unstable focus; (II) a stable focus coexisting with an unstable limit cycle (LC); (III) a stable node; (IV) an unstable focus coexisting with a stable LC; (V) a chaotic dynamics. The green solid line separates the regions with a stable node (III) and a stable focus (II). The blue solid (dashed) curve is a line of super-critical (sub-critical) Hopf bifurcations (HBs), and the red one of saddle-node (SN) bifurcations of LCs. The yellow curve denotes the period doubling (PD) bifurcation lines. In (C), we also report the coherence indicator ρ^(e) (Equation 4) estimated from the network dynamics with N^(e) = 10,000 and N⁽ⁱ⁾ = 2,500. The dashed lines in (A) indicate the parameter cuts we will consider in Figures 4, 5 (black) and Figure 7 (purple), while the open circles in (A,B) denote the set of parameters employed in Figure 11. In the three panels, the inhibitory DC current and the synaptic couplings are fixed to $I_0^{(i)}=I_0^{(e)}/1.02$, $g_0^{(ee)}=0.27$, $g_0^{(ii)}=$0.953939, $g_0^{(ie)}=0.3$, $g_0^{(ei)}=$0.96286; other parameters: (A) K=1,000, ${\Delta }_0^{(ii)}=0.3$, (B) $I_0^{(e)}=$0.001, ${\Delta }_0^{(ii)}=0.3$, (C) K= 1,000, and $I_0^{(e)}=0.1$.

Figure 2

Asynchronous dynamics: Instantaneous population rate R^(e) (R⁽ⁱ⁾) of excitatory (inhibitory) neurons in function of the median in-degree K are shown in (A,B). The effective input currents $I_{eff}^{(e)}$ ($I_{eff}^{(i)}$) given by Equations (13) are reported in (C) and the fluctuations of the input currents $\Delta I_{eff}^{(e)}$ ($\Delta I_{eff}^{(i)}$), as obtained from Equations (20), in (D). Red (blue) color refers to excitatory inhibitory population. The solid continuous lines represent the value obtained by employing the exact MF solutions ${\overline{R}}^{(x)}$ of Equation (8), the dotted (dash-dotted) lines correspond to the first (second) order approximation ${\overline{R}}_0^{(x)}+\epsilon {\overline{R}}_1^{(x)}$ (${\overline{R}}_0^{(x)}+\epsilon {\overline{R}}_1^{(x)}+{\epsilon }^2{\overline{R}}_2^{(x)}$) and the dashed horizontal lines to the zeroth-order one ${\overline{R}}_0^{(x)}$ in (A,B,D), and to $I_a^{(x)}$ in (C) with x = e, i. The circles correspond to data obtained from numerical simulations of N^(e) = N⁽ⁱ⁾ = 10,000 neurons for K < 4,096, N^(e) = N⁽ⁱ⁾ = 20,000 for K = 4,096,8,192 and N^(e) = N⁽ⁱ⁾ = 30,000 for K > 8, 192, averaging the population rates over a window of T = 40 s, after discarding a transient of T = 60 s. The error bars in (A,B) are obtained as the SD (over the time window T) of the population rates, while the average CV of neurons is around 0.15 for all the reported simulations. Synaptic couplings and the ratio between the currents are fixed as stated in sub-section 3.1, other parameters are ${\Delta }_0^{(ii)}=1$, ${\Delta }_0^{(ee)}=2.5$, and $I_0^{(e)}=0.2$. The values of the asymptotic solutions (dashed lines) are : in (A,B) ${\overline{R}}_0^{(e)}=3.18$ Hz and ${\overline{R}}_0^{(i)}=11.28$ Hz, respectively; in (C) $I_a^{(e)}=$ 0.0284 and $I_a^{(i)}\simeq 0.4791$; in (D) $\Delta I_{eff}^{(e)}=$ 0.4623 and $\Delta I_{eff}^{(i)}=$ 0.4593.

Figure 3

Asynchronous dynamics: Probability distribution functions (PDFs) of the total in-degrees $k_j^{(tot)}$ for excitatory (A) and inhibitory (B) active neurons for K = 16, 384. (C,D) Firing rates of the excitatory (inhibitory) neurons $r_j^{(e)}$ ($r_j^{(i)}$) vs. their total in-degrees $k_j^{(tot)}-2K$ symbols refer to K = 1,024 (red), K = 4,096 (blue), and K = 16,384 (green). The inset in (D) is an enlargement of the panel displaying the firing rates over the entire scale $k_j^{(tot)}-2K$. The magenta dashed lines in (C,D) represent the balanced state solution $({\overline{R}}_0^{(e)},{\overline{R}}_0^{(i)})$. (E,F) PDF of the excitatory (inhibitory) firing rates $r_j^{(e)}$ ($r_j^{(i)}$) for K = 16, 384, the solid (dashed) line refers to the MF results ${\overline{R}}^{(x)}$ (${\overline{R}}_0^{(x)}$) with x = e, i. The red (blue) solid line refers to a log-normal fit to the excitatory (inhibitory) PDF with mean 8.8 Hz (17.5 Hz) and SD of 3.8 Hz (2.3 Hz). The parameters are the same as in Figure 1, the firing rates have been estimated by simulating the networks for a total time T_s = 60 s, after discarding a transient T_t = 40 s.

Figure 4

Coherent chaos. (a,b) First (red) λ₁ and second λ₂ (blue) (LEs) for the MF vs. the DC current $I_0^{(e)}$ for the parameter cut corresponding to the dashed black line in Figure 1A. The dashed vertical lines in (a) indicate a super-critical Hopf bifurcation (HB) from a stable focus to periodic COs and the region of the period doubling (PD) cascade. The symbols denote three different types of MF solutions: namely, stable focus (green triangle); periodic oscillations (blue square) and chaotic oscillations (red circle). (c,d) Bifurcation diagrams for the same region obtained by reporting the maximal value of the instantaneous firing rate R^(e) measured from MF simulations. The parameters are the same as in Figure 1, other parameters set as ${\Delta }_0^{(ii)}=0.3$, ${\Delta }_0^{(ee)}=2.0$, K = 1,000.

Figure 5

Different types of collective oscillations(COs). Row (a) refers to the chaotic state observable for $I_0^{(e)}=0.00021$ in the MF denoted by a red circle in Figure 4a; row (b) to the oscillatory state of the MF observable for $I_0^{(e)}=0.0009$ denoted by a blue square in Figure 4a; row (c) to the stable focus for the MF observable for $I_0^{(e)}=0.006$ denoted by a green triangle in Figure 4a. The first column displays the population firing rates vs. time obtained from the network dynamics, the second, the corresponding MF attractors in the planes identified by (R^(e), V^(e)) and (R⁽ⁱ⁾, V⁽ⁱ⁾), the third, the raster plots, and the fourth, the PDFs of the excitatory firing rates $r_j^{(e)}$. Red (blue) color refers to excitatory (inhibitory) populations, the solid vertical lines in column 4 to the mean firing rate and the blue solid line to a fit to a log-normal distribution. Parameters as in Figure 2, apart from ${\Delta }_0^{(ii)}=0.3$, ${\Delta }_0^{(ee)}=2.0$, K = 1,000. For the estimation of the firing rates we employed N^(e) = 40,000 and N⁽ⁱ⁾ = 10,000, while for the raster plots, N^(e) = 10,000 and N⁽ⁱ⁾ = 2,500. The total integration time has been of 120 s after discarding a transient of 80 s.

Figure 6

Pyramidal-interneuron gamma (PING)-like O_P COs. (A) Firing delays Δt between the excitatory population peak and the inhibitory one vs. ${\tau }_m^{(i)}$. Effective mean input currents (Equation 13) (B) and current fluctuations (Equation 20) (C) vs. ${\tau }_m^{(i)}$, the excitatory (inhibitory) population are denoted by red (blue) circles. All the data reported in this study refer to MF simulations. The parameters are $I_0^{(e)}=0.0009$, ${\Delta }_0^{(ii)}=0.3$, ${\Delta }_0^{(ee)}=2.0$, K = 1,000, and ${\tau }_m^{(e)}=20$ ms.

Figure 7

From fluctuation driven to abnormally synchronized oscillations. Firing rates R^(e) (a) and R⁽ⁱ⁾ (b) as a function of $I_0^{(e)}$ for E-I network (circles) and neural mass model (lines) for the parameter cut corresponding to the dashed purple line in Figure 1A. For the neural mass model: solid (dashed) line shows stable (unstable) focus solution ${\overline{R}}^{(e)}$ and ${\overline{R}}^{(i)}$; green dot-dashed lines refer to the extrema of R^(e)(R⁽ⁱ⁾) for the unstable LC present in region (II). The unstable LC emerges at the sub-critical HB for $I_0^{(e)}=74.1709$ separating region (II) from (I), where the focus becomes unstable. Raster plots and PDFs of the excitatory firing rates $r_j^{(e)}$ are reported for specific cases: namely, $I_0^{(e)}=0.128$ (c1,c2), $I_0^{(e)}=1.024$ (d1,d2), and $I_0^{(e)}=100$ (e1,e2). The solid vertical lines in (c2,d2,e2) refer to the mean firing rate. Parameters as in Figure 1, other parameters are set as ${\Delta }_0^{(ii)}=0.3$, ${\Delta }_0^{(ee)}=1.58$, K = 1,000 N^(e) = 10,000, and N⁽ⁱ⁾ = 2,500.

Figure 8

From fluctuation driven to abnormally synchronized oscillations. Coherence indicator ρ (Equation 4) for the whole network of excitatory and inhibitory neurons vs. the excitatory DC current $I_0^{(e)}$ (A) and the median in-degree K (C). Coefficient of variation CV for the whole network vs. $I_0^{(e)}$ (B) and K (D). In (A,C), the symbols refer to different values of the median in-degree:namely, K = 100 (red circles) and K = 4,000 (blue circles). In (B,D), the symbols refer to different excitatory DC currents: namely, $I_0^{(e)}=0.01$ (green circles), $I_0^{(e)}=0.1$ (purple circles), and $I_0^{(e)}=1.0$ (orange circles). Parameters as in Figure 1, other parameters ${\Delta }_0^{(ii)}=0.3$, ${\Delta }_0^{(ee)}=1.58$, N^(e) = 40,000, and N⁽ⁱ⁾ = 10,000.

Figure 9

From quasi-periodicity to frequency locking. (A) Power spectra S(ν) of the mean membrane potential obtained from network simulations. (B) The two fundamental frequencies ν₁(ν₂) vs. $I_0^{(e)}$. (C) Frequency ratio ν₁/ν₂ vs. $I_0^{(e)}$, in the inset ν₁/ν₂ is shown vs. K. (D) Coherence parameter ρ vs. $I_0^{(e)}$, in the inset the corresponding CV is reported. In (B,C), the symbols (solid lines) refer to ν₁ and ν₂ as obtained from the peaks of the power spectra S(ν) for V(t) obtained from the network dynamics (to the two relaxation frequencies ${\nu }_1^R$ and ${\nu }_2^R$ associated to the stable focus solution for the MF). Parameters as in Figure 1, other parameters are set as ${\Delta }_0^{(ii)}=0.3$, ${\Delta }_0^{(ee)}=1.58,N^{(e)}=$80,000, N⁽ⁱ⁾ = 20,000, K = 8,192, and $I_0^{(e)}=$ 0.128 in the inset of (C).

Figure 10

Frequencies and amplitudes of O_F oscillations. The two fundamental frequencies ν₁ and ν₂ vs. $I_0^{(e)}$ (A) and K (C) and the average firing rates vs. $I_0^{(e)}$ (B) and K (D) for the excitatory (red) and inhibitory (blue) populations. In the inset in (C), the effective mean input currents $I_{eff}^{(e)}$ ($I_{eff}^{(i)}$) of the excitatory (inhibitory) population are shown vs. K. The dashed line in (A,C) corresponds to a power law-scaling $\propto {I_0^{(e)}}^{1/2}$ (∝K^1/4) for the frequencies of the COs. The solid red (blue) line in (B,D) denotes the asymptotic MF result ${\overline{R}}^{(e)}$ (${\overline{R}}^{(i)}$). Network (MF) simulations are denoted as stars (circles). The MF data refer to the stable focus, in particular, in (A,C), these are the two relaxation frequencies ${\nu }_1^R$ and ${\nu }_2^R$. Parameters as in Figure 1, other parameters: (A,B) K = 1, 000, ${\Delta }_0^{(ee)}=1.58$, ${\Delta }_0^{(ii)}=0.3$; (C,D) $I_0^{(e)}=0.001$, ${\Delta }_0^{(ee)}=1.3$, ${\Delta }_0^{(ii)}=0.3$; for the network simulations, we employed N^(e) = 80,000 and N⁽ⁱ⁾ = 20,000.

Figure 11

Frequencies and amplitudes of O_P oscillations. COs' frequency ν_CO vs. $I_0^{(e)}$ (A) and K (C) and mean firing rates vs. $I_0^{(e)}$ (B) and K (D) for the excitatory (red) and inhibitory (blue) populations. The dashed line in (A,C) corresponds to a power law-scaling $\propto {I_0^{(e)}}^{1/2}$ (∝K^1/4) for the frequencies. In the inset in (c), the effective mean input currents $I_{eff}^{(e)}$ ($I_{eff}^{(i)}$) of the excitatory (inhibitory) population are shown vs. K. The solid red (blue) line in (B,D) denotes the asymptotic MF result ${\overline{R}}^{(e)}$ (${\overline{R}}^{(i)}$). The data obtained from network (MF) simulations are denoted as stars (circles). The data reported in (A–D) refer to the open circles in Figures 1A,B, respectively. For network simulations, we employed N^(e) = 80,000 and N⁽ⁱ⁾ = 20,000.