# A probabilistic, distributed, recursive mechanism for decision-making in the brain

## Fig 9

The mean number of ISIs to decision in continuous-time spike-trains is equivalent to the mean number observations to decision in discrete time.

*C* sensory neurons (input channels; left) produce sequences of ISIs in continuous time with mean *μ*_{*}*n* (red; best tuned to the stimulus) or *μ*_{0}*n* (black; otherwise). The average decision time—between decision initiation (Init) and termination (Term)—is *τ*_{c} in correct trials, as in this diagram. In discrete time it takes an average of 〈*T*〉_{c} vector observations, **x**(*t*) (composed of scalar observations *x*_{i}(*t*), each time step *t*; blue), to make decisions. [20] showed that in the minimum input case (when *C* = *N*), the mean number of ISIs in the most active channel (red) used by a general, continuous-time, spike-based instance of the MSPRT, approximately equal the mean number of observations, 〈*T*〉_{c}, required by the simpler, discrete-time MSPRT (here 7 in both cases), which carries to our identically-performing rMSPRT; this is true under equal input channel statistics (*μ*_{*}, *μ*_{0}, *σ*_{*}, *σ*_{0}), data distributions (*e.g.* all lognormal), number of alternatives, *N*, and error rate, *ϵ*. This all implies that, if we add 0.5—the expected number of ISIs from decision initiation to a first spike—to 〈*T*〉_{c}, and multiply this by the minimum mean ISI, *μ*_{*}*n* (of fastest firing channel), this approximately equals *τ*_{c}, hence Eq 14; conversely, in error trials we use *μ*_{0} and 〈*T*〉_{e}, to get *τ*_{e} (Eq 15).