Kernel Width Optimization in the Spike-rate ... - Hideaki Shimazaki

Neural Coding 2007 Abstract p120-123, Montevideo, Uruguay

Kernel Width Optimization in the Spike-rate Estimation Hideaki Shimazaki Department of Physics, Kyoto University Kyoto 606-8502, Japan [email protected] Shigeru Shinomoto Department of Physics, Kyoto University Kyoto 606-8502, Japan [email protected]

ABSTRACT A classical tool for estimating the neuronal spike rate is a peri-stimulus time histogram (PSTH) constructed from spike sequences aligned at the onset of a stimulus repeatedly applied to an animal. We have recently established a method for selecting the bin size, so that the PSTH best represents the unknown underlying rate [1]. The goodness of the fit we adopted as the optimization principle is minimizing the mean integrated squared error (MISE) between the ˆt, underlying rate λt and the PSTH λ Z MISE = a

b

ˆ t )2 dt, E(λt − λ

(1)

where E refers to the expectation with respect to the spike generation process under a given time-dependent rate λt . The method allows us to minimize the MISE from spike count statistics alone, without knowing the underlying rate. ˆ t and suggest a method to select the In this contribution, we consider a kernel rate estimator as λ width of a kernel under the MISE criterion. Generally, the cross-validation method is applicable to the least squares minimization [2, 3]. Here, we estimate the MISE fully utilizing the Poissonian nature of spikes, as we have done in the PSTH optimization. The Poissonian assumption holds in the limit of large number of trials, because spikes repeatedly recorded from a single neuron under identical experimental conditions are in the majority mutually independent. For a small number of spike sequences generated from modestly fluctuating underlying rate, the optimal kernel width may become comparable to the observation period. This phenomenon, similar to what we have observed in the PSTH optimization [4], indicates that more experimental trials are needed if one wishes to uncover the time-dependent rate. We also construct a method for estimating the number of additional experimental trials needed to analyze the data with the resolution we deem sufficient. Selection of the Kernel Width We consider independently and identically obtained n spike sequences, which contain N spikes as a whole. Due to the general limit theorem for the sum of independent point processes, a superposition of the spike sequences can be regarded as being drawn from P a time-dependent Poisson point process. We define the superimposed sequence as xt = n−1 N i=1 δ (t − ti ), where ti is the timing of the ith spike. δ(t) is the Dirac delta function. ˆ t is constructed by smoothing the point process by a kernel kw (t) of the bandwidth An estimator λ R ˆ w: λt = kw (t − s)xs ds. We wish to obtain a kernel function that minimizes the MISE (Eq. 1). The integrand of the ˆt + E λ ˆ 2 . Since the first component does not MISE is decomposed into three parts: λ2t − 2λt E λ t depend on the choice of a kernel, we subtract it from the MISE and define a cost function as a

Table 1: A Method for Selecting a Bandwidth of a Kernel Function (i) (ii)

(iii)

Superimpose all the n spike sequences. Obtain a series of spike times {ti }N i=1 in [a, b]. N is the total number of spikes. Compute the cost function of a kernel kw (t) as 4 X 1 X Cˆn (w) = − 2 kw (ti − tj ) + 2 ψw,a,b (ti − tj ) , n n i