Stochastic dynamics of a neural field lattice model with state dependent nonlinear noise

Receptive fields (RF) in the visual cortex can change their size depending on the state of the individual. This reflects a changing visual resolution according to different demands on information processing during drowsiness. So far, however, the possible mechanisms that underlie these size changes have not been tested rigorously. Only qualitatively has it been suggested that state-dependent lateral geniculate nucleus (LGN) firing patterns (burst versus tonic firing) are mainly responsible for the observed cortical receptive field restructuring. Here, we employ a neural field approach to describe the changes of cortical RF properties analytically. Expressions to describe the spatiotemporal receptive fields are given for pure feedforward networks. The model predicts that visual latencies increase nonlinearly with the distance of the stimulus location from the RF center. RF restructuring effects are faithfully reproduced. Despite the changing RF sizes, the model demonstrates that the width of the spatial membrane potential profile (as measured by the variance σ of a gaussian) remains constant in cortex. In contrast, it is shown for recurrent networks that both the RF width and the width of the membrane potential profile generically depend on time and can even increase if lateral cortical excitatory connections extend further than fibers from LGN to cortex. In order to differentiate between a feedforward and a recurrent mechanism causing the experimental RF changes, we fitted the data to the analytically derived point-spread functions. Results of the fits provide estimates for model parameters consistent with the literature data and support the hypothesis that the observed RF sharpening is indeed mainly driven by input from LGN, not by recurrent intracortical connections.

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Stochastic dynamics of snow avalanche occurrence by superposition of Poisson processes

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2012.0396 ◽

2012 ◽

Vol 468 (2148) ◽

pp. 4193-4208 ◽

Cited By ~ 7

Author(s):

Paolo Perona ◽

Edoardo Daly ◽

Benoît Crouzy ◽

Amilcare Porporato

Keyword(s):

Snow Depth ◽

Stochastic Dynamics ◽

Probability Distributions ◽

Poisson Processes ◽

Entropy Function ◽

Constant Frequency ◽

State Variable ◽

Modelling Framework ◽

State Dependent ◽

Deterministic Trajectories

We study the dynamics of systems with deterministic trajectories randomly forced by instantaneous discontinuous jumps occurring according to two different compound Poisson processes. One process, with constant frequency, causes instantaneous positive random increments, whereas the second process has a state-dependent frequency and describes negative jumps that force the system to restart from zero (renewal jumps). We obtain the probability distributions of the state variable and the magnitude and intertimes of the jumps to zero. This modelling framework is used to describe snow-depth dynamics on mountain hillsides, where the positive jumps represent snowfall events, whereas the jumps to zero describe avalanches. The probability distributions of snow depth, together with the statistics of avalanche magnitude and occurrence, are used to explain the correlation between avalanche occurrence and snowfall as a function of hydrologic, terrain slope and aspect parameters. This information is synthesized into a ‘prediction entropy’ function that gives the level of confidence of avalanche occurrence prediction in relation to terrain properties.

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