Optimal responsiveness and information flow in networks of heterogeneous neurons

Abstract

Cerebral cortex is characterized by a strong neuron-to-neuron heterogeneity, but it is unclear what consequences this may have for cortical computations, while most computational models consider networks of identical units. Here, we study network models of spiking neurons endowed with heterogeneity, that we treat independently for excitatory and inhibitory neurons. We find that heterogeneous networks are generally more responsive, with an optimal responsiveness occurring for levels of heterogeneity found experimentally in different published datasets, for both excitatory and inhibitory neurons. To investigate the underlying mechanisms, we introduce a mean-field model of heterogeneous networks. This mean-field model captures optimal responsiveness and suggests that it is related to the stability of the spontaneous asynchronous state. The mean-field model also predicts that new dynamical states can emerge from heterogeneity, a prediction which is confirmed by network simulations. Finally we show that heterogeneous networks maximise the information flow in large-scale networks, through recurrent connections. We conclude that neuronal heterogeneity confers different responsiveness to neural networks, which should be taken into account to investigate their information processing capabilities.

Figure 1

Inhibitory neuron heterogeneity optimizes network responsiveness. (a) An afferent excitatory input (Poissonian spike train at a time-dependent frequency ${\nu }_{\mathit{ext}}(t)$, Input) was submitted to a network of excitatory and inhibitory neurons, whose population spiking activity (amount of spikes per time unit, Output) is measured. The excess of activity in response to the input (blue area) measures the Responsiveness R of the network to the external stimulation. (b) Histograms of the resting potential $e_L=E_L/\overline{E_L}$ of excitatoy (inhibitory) neurons in blue (red), top (low) row. Cells originate from the adult human brain, cortical layers 5/6 and 2/3, and mouse cortical layers 6 and 2/3 (Allen Brain Atlas). The continuous line is a Gaussian distribution with the same standard deviation as measured from the data. (c) Network activity in response to a time varying input ${\nu }_{\mathit{ext}}(t)$ of amplitude $A=1$Hz and baseline value ${\nu }_0=1.5$Hz (see “Methods” section). Excitatory neurons population rate, normalised to pre-stimulus baseline activity, (top row, blue line) and the corresponding raster plot (bottom row), i.e. spiking times of excitatory (inhibitory) neurons marked with blue (red) dots. Panel (c) corresponds to an homogeneous network ${\sigma }_I=0$ , panel (d) to ${\sigma }_I=0.1$ and panel (e) to ${\sigma }_I=0.15$ (in these panels (c)–(e) ${\sigma }_E=0$). (f) The evoked response R is reported in function of the heterogeneity of inhibitory (excitatory) neurons $\sigma ={\sigma }_I$ ($\sigma ={\sigma }_E)$, red (blue) dots (squares). Error bars are estimated as the standard deviation over 20 different realisations. Continuous lines report the prediction based on the mean field model (see main text). In the inset, R is reported in function of the spontaneous excitatory firing rate $r_E$ without the stimulus (average over 10 s). Red dots are obtained by varying ${\sigma }_I$ (same data as the main panel). Green dots show R in function of $r_E$ for an homogeneous network (${\sigma }_I={\sigma }_E=0$) and different values of the average resting potential of inhibitory neurons $\overline{E_L^I}$.

Figure 2

Excitatory and inhibitory heterogeneity determines network responsiveness. (a) Responsiveness R as a function of heterogeneity of excitatory (${\sigma }_E$) and inhibitory (${\sigma }_I$) neurons for networks displaying relatively high levels of spontaneous activity (input baseline ${\nu }_0=1.5Hz$ and input amplitude $A=1$ Hz). (b) Responsiveness R for networks with a lower level of spontaneous activity (input baseline ${\nu }_0=0.4Hz$ and input amplitude $A=0.6$ Hz). Different markers correspond to heterogeneity values estimated from Human cortex layer 2, 3 (white circle), Human cortex layer 5, 6 (grey circle), mouse cortex layer 2/3 (grey star) and mouse cortex layer 6 (white diamond) estimated from Allen Brain database, see Fig. 1.

Figure 3

Enhanced responsiveness corresponds to regions closer to instability. (a) The second largest stability Lyapunov exponent $\lambda$ (real part) of asynchronous dynamics as a function of the heterogeneity of inhibitory neurons (${\sigma }_I$). Different colors indicate different parameters of the baseline external drive and the strength of excitatory-excitatory quantal conductance (${\nu }_0,Q_{\mathit{EE}}$), i.e. black (1.5 Hz, 1.5 nS), red (3 Hz, 1.5 nS), blue (2 Hz, 1.5 nS) and orange (3 Hz, 1.65 nS). Symbols are located at the value of ${\sigma }_I$ for which the responsiveness R is maximum (same color code as continuous line). Different symbols indicate different amplitudes A of the input, diamond ($A=0.1$ Hz), star ($A=0.5$ Hz) and dot ($A=1$ Hz). (b) The second largest stability Lyapunov exponent $\lambda$ of asynchronous dynamics as a function of the heterogeneity of inhibitory neurons (${\sigma }_I$) and the strength of excitatory-excitatory quantal conductance $Q_{\mathit{EE}}$ for a baseline external drive ${\nu }_0=1.5$ Hz. The dotted (diamond) line is the responsiveness R for an input amplitude $A=0.1$ Hz ($A=1Hz$) and $Q_{\mathit{EE}}=1.5$ nS (as in Figs. 1 and 2). Such responsiveness has been properly rescaled on the y-axes (i.e. multiplied by an ad-hoc factor) in order to fit in the image.

Figure 4

Heterogeneous networks admit more dynamical regimes compared to homogeneous networks. (a) Inhibitory neurons population stationary firing rate $r_I^{\ast }$ in function of ${\sigma }_I$ for $\overline{E_L^I}=-70$ mV. Black line indicates stable ($\lambda <0$) asynchronous state, dashed blue line indicates unstable ($\lambda >0$) asynchronous state where a limit cycle appears (red lines indicates maximum and minimum value of $r_I$ in time). (b) The second stability Lyapunov exponent $\lambda$ (real part) of the asynchronous state as a function of the average resting potential of inhibitory neurons $\overline{E_L^I}$ and their standard deviation ${\sigma }_I$. Whenever $\lambda >0$ the asynchronous state is unstable and a stable limit cycle appears. In direct network simulations we observe sparsely synchronous oscillations (see raster plots in panel (c) where we use $\overline{E_L^I}=-70$ mV and ${\sigma }_I=0,0.12,0.2$ from top to bottom). In these simulations $Q_{\mathit{EE}}=1.53$ nS.

Figure 5

Enhanced activity propagation in large-scale heterogeneous networks. (a) Two dimensional lattice of connected mean-field models. The connectivity between excitatory (inhibitory) cells is drawn from a Gaussian distribution with standard deviation of length $l_{\mathit{exc}}=2$ mm ($l_{\mathit{inh}}=1$ mm), see blue (red) curves. (b) Response of the system (firing rate of excitatory neurons) following a Gaussian afferent stimulus (see “Methods”). The upper row stands for locally homogeneous networks (${\sigma }_I=0$) while lower row for locally heterogeneous networks (${\sigma }_I=0.15$). (c) Spatio–temporal profile of excitatory neurons firing rate normalised by its maximum in space and time (${\sigma }_I=0$). (d) Same as (c) but for ${\sigma }_I=0.15$. (e) Normalised firing rate in function of the distance d from stimulus onset at a specific time [see dashed white line in panel (d)]. Different colors stand for different levels of heterogeneity (see the legend inside the panel). The dashed red line shows a fit of the exponential decay of activity found in the heterogeneous system (red line, ${\sigma }_I=0.15$), with ${\lambda }_d\sim 3.5$ mm.