Fox and Lu introduced a Langevin framework for discrete-time stochastic models of randomly gated ion channels such as the Hodgkin-Huxley (HH) system. They derived a Fokker-Planck equation with state-dependent diffusion tensor [Formula: see text] and suggested a Langevin formulation with noise coefficient matrix [Formula: see text] such that SS[Formula: see text]. Subsequently, several authors introduced a variety of Langevin equations for the HH system. In this letter, we present a natural 14-dimensional dynamics for the HH system in which each directed edge in the ion channel state transition graph acts as an independent noise source, leading to a 14 [Formula: see text] 28 noise coefficient matrix [Formula: see text]. We show that (1) the corresponding 14D system of ordinary differential equations is consistent with the classical 4D representation of the HH system; (2) the 14D representation leads to a noise coefficient matrix [Formula: see text] that can be obtained cheaply on each time step, without requiring a matrix decomposition; (3) sample trajectories of the 14D representation are pathwise equivalent to trajectories of Fox and Lu's system, as well as trajectories of several existing Langevin models; (4) our 14D representation (and those equivalent to it) gives the most accurate interspike interval distribution, not only with respect to moments but under both the [Formula: see text] and [Formula: see text] metric-space norms; and (5) the 14D representation gives an approximation to exact Markov chain simulations that are as fast and as efficient as all equivalent models. Our approach goes beyond existing models, in that it supports a stochastic shielding decomposition that dramatically simplifies [Formula: see text] with minimal loss of accuracy under both voltage- and current-clamp conditions.
Neurons in the dorsolateral prefrontal cortex (dlPFC) and posterior parietal cortex (PPC) are activated by different cognitive tasks and respond differently to the same stimuli depending on task. The conjunctive representations of multiple tasks in nonlinear fashion in single neuron activity, is known as nonlinear mixed selectivity (NMS). Here, we compared NMS in a working memory task in areas 8a and 46 of the dlPFC and 7a and lateral intraparietal cortex (LIP) of the PPC in macaque monkeys. NMS neurons were more frequent in dlPFC than in PPC and this was attributed to more cells gaining selectivity in the course of a trial. Additionally, in our task, the subjects’ behavioral performance improved within a behavioral session as they learned the session-specific statistics of the task. The magnitude of NMS in the dlPFC also increased as a function of time within a single session. On the other hand, we observed minimal rotation of population responses and no appreciable differences in NMS between correct and error trials in either area. Our results provide direct evidence demonstrating a specialization in NMS between dlPFC and PPC and reveal mechanisms of neural selectivity in areas recruited in working memory tasks.
In this paper, a new class of five parameter gamma-exponentiated or generalized modified Weibull (GEMW) distribution which includes exponential, Rayleigh, Weibull, modified Weibull, exponentiated Weibull, exponentiated exponential, exponentiated modified Weibull, exponentiated modified exponential, gamma-exponentiated exponential, gammaexponentiated Rayleigh, gamma-modified Weibull, gamma-modified exponential, gamma-Weibull, gamma-Rayleigh and gamma-exponential distributions as special cases is proposed and studied. Mathematical properties of this new class of distributions including moments, mean deviations, Bonferroni and Lorenz curves, distribution of order statistics and Renyi entropy are presented. Maximum likelihood estimation technique is used to estimate the model parameters and applications to real data sets presented in order to illustrate the usefulness of this new class of distributions and its sub-models.
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