Abstract. We review the recent proposal that the most fascinating brain properties are related to the fact that it always stays close to a second order phase transition. In such conditions, the collective of neuronal groups can reliably generate robust and flexible behavior, because it is known that at the critical point there is the largest abundance of metastable states to choose from. Here we review the motivation, arguments and recent results, as well as further implications of this view of the functioning brain.
A new computationally simple, speedy and accurate method is proposed to construct first-passage-time probability density functions for Gauss–Markov processes through time-dependent boundaries, both for fixed and for random initial states. Some applications to Brownian motion and to the Brownian bridge are then provided together with a comparison with some computational results by Durbin and by Daniels. Various closed-form results are also obtained for classes of boundaries that are intimately related to certain symmetries of the processes considered.
Two diffusion processes with multiplicative noise, able to model the changes in the neuronal membrane depolarization between two consecutive spikes of a single neuron, are considered and compared. The processes have the same deterministic part but different stochastic components. The differences in the state-dependent variabilities, their asymptotic distributions, and the properties of the first-passage time across a constant threshold are investigated. Closed form expressions for the mean of the first-passage time of both processes are derived and applied to determine the role played by the parameters involved in the model. It is shown that for some values of the input parameters, the higher variability, given by the second moment, does not imply shorter mean firstpassage time. The reason for that can be found in the complete shape of the stationary distribution of the two processes. Applications outside neuroscience are also mentioned.
The aim of this paper is the construction of stochastic versions for some fractional Gompertz curves. To do this, we first study a class of linear fractional-integral stochastic equations, proving existence and uniqueness of a Gaussian solution. Such kinds of equations are then used to construct fractional stochastic Gompertz models. Finally, a new fractional Gompertz model, based on the previous two, is introduced and a stochastic version of it is provided.
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