Cross-correlation--response relation for spike-driven neurons

We consider two spiking neurons, with one neuron driving the other. From a statistical point of view, the relationship between the two neurons can be quantified by (i) the cross-correlation $C(t,t^{\prime})$ of the two spike trains, a measure of similarity of spontaneous fluctuations, and (ii) the linear response $K(t,t^{\prime})$ of the driven neuron's rate to a rate perturbation of the driving neuron. Connecting the two statistics, we analytically derive a simple cross-correlation–response relation (CRR), $C(t,t^{\prime})=K(t,t^{\prime})\lambda(t^{\prime})$, for an arbitrary model of the driven neuron, provided that the driving neuron's spiking can be approximated by an inhomogeneous Poisson process. Here $\lambda(t)$ is the time-dependent firing rate of the driving neuron. For temporally colored (non-Poissonian) input spikes, we find $C(t,t^{\prime})\approx\int ds\,K(t,s)C_{\text{in}}(s,t^{\prime})$, where the input-spikes' autocorrelation $C_{\text{in}}(t,t^{\prime})\rightarrow\lambda(t)\delta(t-t^{\prime})$ in the Poissonian limit, and the approximation holds for neurons that are weakly sensitive to input autocorrelations. The CRRs can be considered as analogues of the Furutsu-Novikov theorem (FNT) [1,2], which has recently been applied in neuroscience [3,4,5]. However, while for the FNT the input has to be a Gaussian process, the CRRs apply to shot-noise input, taking properly into account that neurons communicate via spikes. The relations are numerically tested [Fig.(a-b)] in frequency space for a Poisson-driven leaky integrate-and-fire neuron. As an example, we consider remote controlling one neuron in a recurrent network by stimulating another one [Fig.(d,e)]. This task requires knowledge of the second neuron's response function to modulations of the first neuron's rate, which in turn requires a large number of trials involving precise stimulation. However, using our CRR and assuming weakly correlation-sensitive neurons, this response function is obtained from the spontaneous input-output cross-correlation [Fig.(f,g)]. From a more theoretical point of view, CRRs can be helpful by constraining the constituents in the theory of neural networks. For example, in a recent cavity-method approach to rate-based neural networks [4] with Gaussian statistics, the Gaussian FNT was used to connect neural cross-correlations and response functions. With the CRRs presented here, such approaches can likely be extended to recurrent networks of spiking neurons.