Extended Data Fig. 6: Multi-mode drift-diffusion model (MMDDM).

a, Directed graph of the MMDDM for a trial with T time steps and N simultaneously recorded neurons. At each time step, the decision variable z depends on external click input (u) and its value in the previous time step. The spike train depends on z and also a time-varying baseline input. The behavioral choice (c) is the sign of the decision variable at the last time step. In this example trial, z reaches the bound, and the encoding weight of z of each neuron changes from w_EA to w_DC. b, The MMDDM is an instance of a state-space model, which consists of a dynamic model governing the probability distributions of the latent states (here, scalar decision variable z) and measurement models specifying the conditional distributions of the emission (here, spike counts y and the rat’s choice c) given the value of the latent states. In the dynamic model, z’s time derivative (ż) is a piecewise linear function. When the absolute value of z is less than the bound height B, the velocity depends on external click input (u) and i.i.d. Gaussian noise (η). When z reaches either -B or B, the time derivative is zero. The input of each click emitted at time τ on z is scaled by the depressive adaptation from previous clicks, parametrized by C(τ), and it is corrupted by i.i.d. multiplicative Gaussian noise ζ with variance σ_s². The parameter σ_s² is one of the three parameters learned during fitting and represents the signal-to-noise of the system. The behavioral choice (c) is the sign of the decision variable at the last time step. The mapping from z to spike train response (y) passes through the softplus nonlinearity h and depends on baseline b and encoding weight w. The encoding weight is either w_EA and w_DC depending on z. The three parameters that are fit in MMDDM consist of the bound height B, the mean μ₀ of starting distribution, and the signal-to-noise of each momentary input. c, The baseline input consists of a cross-trial component, parametrized by smooth temporal basis functions, as shown for an example neuron. d, The spike history filter of the same neuron. e, The post-stimulus filter of the neuron. This filter does not depend on the content of the click train and only depends on the timing of the first click, which is always a simultaneous left and right click. f, The kernel of the same neuron to account for movement anticipation. The kernel does not depend on the actual choice of the animal. g, The psychometric function is well captured across sessions. h, The vector field inferred from real spike trains is confirmed to be similar to that inferred from MMDDM-simulated spike trains for the session “T176_2018_05_03”. i, After fitting the model to each recording session, the learned parameters are used to simulate a data set, using the same number of trials and the same auditory click trains. The simulations are used to fit a new model, the recovery model, starting from randomized parameter values. The encoding weights of the accumulated evidence of the recovery model are compared against the weights used for the simulation (which were learned by fitting to the data) using the coefficient-of-determination metric. j, Consistency in the encoding weights between the training models during five-fold cross-validation. For each session, a coefficient-of-determination was computed for each pair of training models (10 pairs), and the median is included in the histogram. k, Whereas the Poisson distribution requires the mean to be the same as the variance, the negative binomial distribution is a count response model that allows the variance to be larger than the mean μ, with an additional parameter α, the overdispersion parameter, that specifies the variance to be equal to μ + αμ². When the overdispersion parameter is zero, the distribution is equivalent to a Poisson. Fitting the data to varying values of the overdispersion parameter shows that log-likelihood is maximized with a Poisson distribution for the conditional spike count response. Similarly, when the overdispersion parameter was learned from the data, the best-fit values were all close to zero. l, The magnitude of the input after sensory adaptation of each click in a simulated Poisson auditory click train. Based on previous findings²⁴, the adaptation strength (φ) is fixed to 0.001, and the post-adaptation recovery rate (k) to 100. The generative click rate is 40 Hz, as in the behavioral task. m, Sensory adaptation is not critical to the improvement in fit by the MMDDM compared to the single mode DDM. Even without modeling sensory adaptation–by setting φ = 1 and k = 0, such that every click has the same input magnitude–the out-of-sample log-likelihood is reliably improved by the MMDDM compared to the single mode DDM. n, The out-of-sample goodness-of-fit of the PSTH’s is also reliably improved even in the absence of sensory adaptation. m-n, P-values were computed using two-sided sign tests.