Which model can properly describe dynamics and smoothness of firing rate ?

1 Probabilistic model We construct likelihood function and prior distribution. Let observation time T consist of M bins whose width is ∆, assuming that T is the common denominator of ∆ for simplicity. In the condition that N trials give total spike numbers ηm to the mth bin (m = 1, 2, ...,M), likelihood function is defined by p(ηm|λm) ∝ λm m exp(−N∆λm), (1) where λm is the firing rate at the mth bin. Line process model, Gauss model, and Cauchy model, which models we use as the prior distributions, are


Probabilistic model
We construct likelihood function and prior distribution.Let observation time T consist of M bins whose width is ∆, assuming that T is the common denominator of ∆ for simplicity.In the condition that N trials give total spike numbers η m to the mth bin (m = 1, 2, ..., M ), likelihood function is defined by where λ m is the firing rate at the mth bin.Line process model, Gauss model, and Cauchy model, which models we use as the prior distributions, are respectively.Note that the domains of the firing rate {λ} = {λ 1 , λ 2 , ..., λ M } are [0, ∞).The Line process is define by an (M − 1) dimensional state vector {l} = {l 2 , l 3 , ..., l M } ∈ {0, 1} M −1 .Here l m+1 = 0 if λ m+1 continuously evolves from λ m , and l m+1 = 1 if it evolves discontinuously from λ m .Each model contains the hyperparameter β that controls the smoothness of each distribution.The Line process model contains the hyperparameter α that controls the frequency of the firing rate discontinuity.The posterior distribution is where M represents the model as a whole, i.e.M is the Line process model, the Gauss model, or the Cauchy model, assuming that {l} = {l 2 , l 3 , ..., l N } ∈ {0} M −1 in the Gauss model and the Cauchy model.

Firing rate
Since maximizer of posterior marginals(MPM) gives us the estimated firing rate values as λm = arg max λm p(λ m |{η}), we need to calculate the marginal posterior distribution.We define two messages common to three models as , where both the initial values, i.e. µ L (λ 1 ) and µ R (λ N ), are 1.We can write a marginal posterior distribution as where {η} = {η 1 , η 2 , ..., η M }.

Stimulus timings
The marginal posterior distribution of l m gives us the estimated unknown stimulus timing that occurs at the mth bin if p(l m = 1|β, α) > 0.5.The belief propagation gives us the marginal distribution of l m as Figure 1(c) and 1(d) show the estimated stimulus timings using the Line process model in the second experiment.

Marginal likelihood
We estimate the hyperparameters and select the appropriate model to maximize the log marginal likelihood as follows based on empirical Bayes method and model selection, respectively: The log marginal likelihood is where Z posterior (β, α|M) and Z prior (β, α|M) are a normalization constant of a posterior distribution and a normalization constant of a prior distribution, respectively.The message, regarding to a prior distribution, which message is gives us a normalization constant of a prior distribution as Z prior (β, α|M) = ∫ dλ 1 μR (λ 1 |M).The messages, given by eq. ( 6) and eq.( 7), give us a normalization constant of a posterior distribution as Z posterior (β, α|M) = ∫ dλ 1 µ R (λ 1 |M).We set α to 0 in both the Gauss model and the Cauchy model.Table 1 denotes the results in the first experiment and the second experiment, which results imply the Line process model is appropriate to describe the dynamics and the smoothness of the firing rate in both the first experiment and the second experiment.

Figure 1 :
Figure 1: Estimated firing rate values and estimated unknown stimulus timings.(a) and (b) denote the estimated firing rate values in the first experiment and the second experiment, respectively, using six spike trains and the Line process model.True firing rate is represented by the dashed line, and the estimated firing rate is by the solid line.(c) and (d) show the estimated unknown stimulus timings in the second experiment using three spike trains and six spike trains, respectively, by use of the Line process model.We express estimated stimulus timings and true stimulus timings using solid line and dashed line, respectively.

Table 1 :
The calculated log marginal likelihood in the first experiment and the second experiment.