We show that MID is a maximum-likelihood estimator for the parameters of a Iinear—nonlinear—Poisson (LNP) model, and that the empirical single-spike information corresponds to the normalized log-likelihood under a Poisson model .

However, conditionally independent spiking is also the fundamental assumption underlying the Poisson model and, as we have shown, the standard MID estimator (based on the KL-divergence between histograms) is mathematically identical to the maximum likelihood estimator for an LNP model with piecewise constant nonlinearity.

In fact, the quantity —£lnp(90, D) can be considered an estimate for the marginal entropy of the response distribution, H (r) = —Z p(r) log p(r), since it is the average log-probability of the response under a Poisson model , independent of the stimulus.

Spike-history dependencies can influence the single-bin spike count distribu-tion—for example, a Bernoulli model is more accurate than a Poisson model when the bin size is smaller than or equal to the refractory period, since the Poisson model assigns positive probability to the event of having 2 2 two spikes in a single bin.

Under the discrete-time inhomogeneous Poisson model considered above, spikes are modeled as conditionally independent given the stimulus, and the spike count in a discrete time bin has a Poisson distribution.

The Bernoulli and discrete-time Poisson models approach the same limiting Poisson process as the bin size (and single-bin spike probability) approaches zero while the average spike rate remains constant.

Specifically, the empirical single-spike information is equal to the log-likelihood ratio between an inhomogeneous and homogeneous Poisson model of the repeat data (normalized by spike count):

Where XML denotes the maximum-likelihood or plugin estimate of the time-varying spike rate (i.e., the PSTH itself), 2: is the mean spike rate across time, and £(9c;r) denotes the log-likeli-hood of the repeat data r under a Poisson model With time-varying rate 2v.

Thus, even When estimated from raster data, Iss is equal to the difference between Poisson log-likelihoods under an inhomogeneous (rate-varying) and a homogeneous (constant rate) Poisson model , diVided by spike count (see also [57] ).

For the analytical treatment, the neurons in the output and lateral layers were modeled with a linear Poisson model .

In the main text, we performed all simulations With a linear Poisson model for analytical purposes, although we also confirmed those results With a conductance-based LIF model (81 Fig).

In the LIF model, synaptic weights develop in a manner similar to that for the linear Poisson model , although change occurs more rapidly (Fig IB, SIA Fig).

Correction of the modulation index for deviations from the Poisson model

The deviation of the spike train from the Poisson model results in an incorrect estimation of the modulation indeX.

(3) From this rate function, multiple new Poisson-like spike trains are generated, including the given deviations from the Poisson model .

In real neurons, there are deviations from the Poisson model .

The general rate function model could be eXpanded to accommodate for other deviations from the Poisson model , such as relative refractory period and bursting activity; i.e., an elevated firing probability immediately after the refractory period.

However, real neurons deviate from the Poisson model , primarily as a result of the refractory period.