Theory of oscillatory firing induced by spatially correlated noise and delayed inhibitory feedback

A network of leaky integrate-and-fire neurons with global inhibitory feedback and under the influence of spatially correlated noise is studied. We calculate the spectral statistics of the network (power spectrum of the population activity, cross spectrum between spike trains of different neurons) as well as of a single neuron (power spectrum of spike train, cross spectrum between external noise and spike train) within the network. As shown by comparison with numerical simulations, our theory works well for arbitrary network size if the feedback is weak and the amount of external noise does not exceed that of the internal noise. By means of our analytical results we discuss the quality of the correlation-induced oscillation in a large network as a function of the transmission delay and the internal noise intensity. It is shown that the strongest oscillation is obtained in a system with zero internal noise and adiabatically long delay (i.e., the delay period is longer than any other time scale in the system). For a neuron with a strong intrinsic frequency, the oscillation becomes strongly anharmonic in the case of a long delay time. We also discuss briefly the kind of synchrony introduced by the feedback-induced oscillation.

Figure 1

The network model. Pyramidal cells (circles) receive correlated and uncorrelated external stimuli as well as inhibitory feedback of their spike trains. This feedback consists of the sum of all spike trains convoluted by an $\alpha$ function with time constant ${\tau }_S$ and delayed by a constant ${\tau }_D$ corresponding to the finite axonal transmission time.

Figure 2

Power spectrum of a single neuron in a network of $N=100$ neurons for $c=0$ (squares, dashed lines) and $c=1$ (circles, solid lines); simulations (symbols) compared to theory Eq. (32). Parameters are $D=0.12$, $G=-1.2$, $\mu =0.8$, ${\tau }_R=0.1$, $D_E=0.08$, ${\tau }_S=0.5$, and ${\tau }_D=1$.

Figure 3

Spectral measures obtained by simulations (symbols) compared to theory (lines) for $c=0$ (squares, dashed lines) and $c=1$ (circles, solid lines) and different system size $N$; from top to bottom (with theoretical expressions given in brackets): power spectrum of a single neuron [Eq. (32)], real part and imaginary part of the cross spectrum between the common part of the external noise and the spike train of a single neuron [Eq. (33)], power spectrum of the feedback kernel [Eq. (36)], spectrum of the population activity [Eq. (35)], and cross spectrum between spike trains from distinct neurons [Eq. (34)]. Network size is $N=1$ (left column), $N=2$ (mid column), and $N=100$ (right column). Parameters are $\mu =0.8$, $D=0.12$, $D_E=0.08$, $G=-0.5$, ${\tau }_R=0.1$, ${\tau }_S=0.5$, and ${\tau }_D=1$.

Figure 4

Improved theory for the cross spectrum for $\mu =0.8$, $D=0.12$, $D_E=0.08$, $G=-0.5$, ${\tau }_R=0.1$, ${\tau }_S=0.5$, and ${\tau }_D=1$. Upper panel: The full simulation result (black) is compared to the purely theoretical result Eq. (34). This is a log plot of the data shown in Fig. 3 (right panel in lowest row). Lower panel: The same simulation result is compared to Eq. (38) where we have used the numerically computed spike train cross spectrum of two uncoupled neurons driven by currents given in Eq. (37) with ${\mu }^{\prime }=0.6234$ (incorporating the static part of the feedback for $G=-0.5$). Since the latter cross spectrum was estimated by another simulation (of only two neurons), the corresponding “theory” curve is not smooth. Note the much better agreement in the lower panel.

Figure 5

Improved theory for the cross spectrum for $\mu =0.8$, $D=0.12$, $D_E=0.08$, $G=-1.0$, ${\tau }_R=0.1$, ${\tau }_S=0.5$, and ${\tau }_D=6.324$ (feedback strength and delay have been increased compared to the parameters in Fig. 4). Upper panel: The full simulation result (black) is compared to the purely theoretical result Eq. (34). Lower panel: The same simulation result is compared to Eq. (38) where we have used Eq. (42). The latter cross spectrum was estimated by another network simulation using the parameters above except for the synaptic decay time which we set ${\tau }_S={10}^3$.

Figure 6

Spike train power spectrum (a) and spectrum of the network activity (b) for different delay time ${\tau }_D$ and a subthreshold bias current $\mu =0.8$. The data were obtained by numerical simulation of a network with $N=100$. Other parameters: $G=-1.0$, $D=0.12$, $D_E=0.08$, ${\tau }_R=0.1$, and ${\tau }_S=0.5$.

Figure 7

Power spectrum of the population activity for a large delay time ${\tau }_D=20$. Symbols are the result of simulations that agree well with theory [solid line, Eq. (35)] except for the peak height which is overestimated by the theory. Other parameters are as in Fig. 6.

Figure 8

Oscillation frequency (a) and spectral coherence (b) of the first peak in the power spectrum of the network activity vs delay time ${\tau }_D$. The width of the first peak in panel (a) is indicated by the grey area and is defined by $S\left(\omega \right)>S\left({\omega }_{\max }\right)∕2$. Theoretical curves have been obtained by numerical evaluation of the analytical expression Eq. (35). Simulation results were obtained by smoothing the simulation spectra sufficiently by running averages (length of average depends on system’s parameters) and estimating the oscillation frequency and peak width by eye. The dashed line shows the angular frequency that corresponds to twice the delay time $(\pi ∕{\tau }_D)$—this frequency is approached by the oscillation frequency at large delay as seen in the log-log plot in the inset. Parameters are $D=0.12$, $G=-1.0$, $\mu =0.8$, ${\tau }_R=0.1$, $D_E=0.08$, and ${\tau }_S=0.5$.

Figure 9

Spike train power spectrum (a) and spectrum of the network activity (b) for different intensities of the internal noise and subthreshold bias current $\mu =0.8$, obtained by numerical simulation of a network with $N=100$. Other parameters: $D_E=0.08$, $G=-1.0$, ${\tau }_D=1$, ${\tau }_R=0.1$, and ${\tau }_S=0.5$.

Figure 10

Oscillation frequency $\left({\omega }_{\max }\right)$ and width of the first peak [peak area is grey and is defined by $S\left(\omega \right)<S\left({\omega }_{\max }\right)∕2$] in the spectrum of the population activity as a function of the internal noise intensity. Theoretical curves have been obtained by numerical evaluation of the analytical expression Eq. (35). Simulation results were obtained by smoothing the simulation spectra sufficiently by running averages (length of average depends on system’s parameters) and estimating the oscillation frequency and peak width by eye. Parameters are $D_E=0.08$, $G=-1.0$, $\mu =0.8$, ${\tau }_R=0.1$, ${\tau }_D=1$, and ${\tau }_S=0.5$.

Figure 11

Spike train power spectrum (a) and spectrum of the network activity (b) for different values of the delay time ${\tau }_D$ and suprathreshold bias current $\mu =1.7$, lower noise and smaller synaptic time scale than in the previous figures. Parameters are $D=0.025$, $D_E=0.025$, $G=-0.5$, ${\tau }_R=0.1$, and ${\tau }_S=0.05$.

Figure 12

Auto- and cross-correlation functions of single neuron spike trains for a large network with $N=100$. (a) Cross correlation of two spike trains from distinct neurons. Simulation (grey circles) are compared to theory [solid line, numerically calculated inverse Fourier transform of Eq. (34)). The cross correlation shows minima and maxima at multiples of the delay time ${\tau }_D=6.324$. (b) Autocorrelation function. Simulations (grey circles) vs theory [solid line, numerically calculated inverse Fourier transform of Eq. (32)]. Again minima and maxima are observed at multiples of ${\tau }_D=6.324$. (c) Comparison of the simulation data for cross- (solid line) and autocorrelation functions. Note that both differ only for small $\tau$ but agree well at the oscillation-induced peaks at multiples of the delay time ${\tau }_D=6.324$.