Abstract
The mutual information between stimulus and spike-train response is commonly used to monitor neural coding efficiency, but neuronal computation broadly conceived requires more refined and targeted information measures of input-output joint processes. A first step toward that larger goal is to develop information measures for individual output processes, including information generation (entropy rate), stored information (statistical complexity), predictable information (excess entropy), and active information accumulation (bound information rate). We calculate these for spike trains generated by a variety of noise-driven integrate-and-fire neurons as a function of time resolution and for alternating renewal processes. We show that their time-resolution dependence reveals coarse-grained structural properties of interspike interval statistics; e.g., τ-entropy rates that diverge less quickly than the firing rate indicated by interspike interval correlations. We also find evidence that the excess entropy and regularized statistical complexity of different types of integrate-and-fire neurons are universal in the continuous-time limit in the sense that they do not depend on mechanism details. This suggests a surprising simplicity in the spike trains generated by these model neurons. Interestingly, neurons with gamma-distributed ISIs and neurons whose spike trains are alternating renewal processes do not fall into the same universality class. These results lead to two conclusions. First, the dependence of information measures on time resolution reveals mechanistic details about spike train generation. Second, information measures can be used as model selection tools for analyzing spike train processes.
Highlights
Substituting in Equations (3), (4), and (5), we find that the excess entropy E tends to: FIGURE 2 | ǫ-Machines of processes generated by Poisson neurons and by integrate-and-fire neurons: (A) The ǫ-machine for a Poisson process. (B) The ǫ-machine for an eventually Poisson process; i.e., a Poisson neuron with a refractory period of length nt. (C) The ǫ-machine for a generic renewal process—the not eventually -Poisson process of Marzen and Crutchfield (2015); i.e., the process generated by noise-driven integrate-and-fire neurons
We explored the scaling properties of a variety of informationtheoretic quantities associated with two classes of spiking neural models: renewal processes and alternating renewal processes
We found that information generation and stored information both diverge logarithmically with decreasing time resolution for both types of spiking models, whereas the predictable information and active information accumulation limit to a constant
Summary
Despite a half century of concerted effort (Mackay and McCulloch, 1952), neuroscientists continue to debate the relevant timescales of neuronal communication as well as the basic coding schemes at work in the cortex, even in early sensory processing regions of the brain thought to be dominated by feedforward pathways (Softky and Koch, 1993; Bell et al, 1995; Shadlen and Newsome, 1995; Stevens and Zador, 1998; Destexhe et al, 2003; DeWeese and Zador, 2006; Jacobs et al, 2009; Koepsell et al, 2010; London et al, 2010). Compounding the need for better theoretical tools, measurement techniques will soon amass enough data to allow serious study of neuronal communication at fine time resolutions and across large populations (Alivisatos et al, 2012). Many single-neuron models generate neural spike trains that are renewal processes (Gerstner and Kistler, 2002) Starting from this observation, we use recent results (Marzen and Crutchfield, 2015) to determine how information measures scale in the small time-resolution limit. We extend the previous analyses to structurally more complex, alternating renewal processes and analyze the time-resolution scaling of their information measures This yields important clues as to which scaling results apply more generally. This reveals a new kind of universality in which the information measures’ scaling is independent of detailed spiking mechanisms.
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