Mario Annunziato

Spike-Driven Synaptic Plasticity: Theory, Simulation, VLSI Implementation

We present a model for spike-driven dynamics of a plastic synapse, suited for a VLSI implementation. The synaptic device behaves as a capacitor on short timescales and preserves the memory of two stable states (efficacies) on long timescales. The transitions (LTP/LTD) are stochastic because both the number and the distribution of neural spikes in any finite (stimulation) interval fluctuate, even at fixed pre- and postsynaptic spike rates. The dynamics of the single synapse is studied analytically by extending the solution to a classic problem in queuing theory (Takcs process). The model of the synapse is implemented in a VLSI and consists of only 18 transistors. It is also directly simulated. The simulations indicate that LTP/LTD probabilities versus rates are robust to fluctuations of the electronic parameters in a wide range of rates. The solutions for these probabilities are in very good agreement with both the simulations and measurements. Moreover, the probabilities are readily manipulable by variations of the chips parameters, even in ranges where they are very small. The tests of the electronic device cover the range from spontaneous activity (34 Hz) to stimulus-driven rates (50 Hz). Low transition probabilities can be maintained in all ranges, even though the intrinsic time constants of the device are short ( 100 ms). Synaptic transitions are triggered by elevated presynaptic rates: for low presynaptic rates, there are essentially no transitions. The synaptic device can preserve its memory for years in the absence of stimulation. Stochasticity of learning is a result of the variability of interspike intervals; noise is a feature of the distributed dynamics of the network. The fact that the synapse is binary on long timescales solves the stability problem of synaptic efficacies in the absence of stimulation. Yet stochastic learning theory ensures that it does not affect the collective behavior of the network, if the transition probabilities are low and LTP is balanced against LTD.

A Fokker-Planck control framework for multidimensional stochastic processes

An efficient framework for the optimal control of probability density functions (PDFs) of multidimensional stochastic processes is presented. This framework is based on the Fokker-Planck equation that governs the time evolution of the PDF of stochastic processes and on tracking objectives of terminal configuration of the desired PDF. The corresponding optimization problems are formulated as a sequence of open-loop optimality systems in a receding-horizon control strategy. Many theoretical results concerning the forward and the optimal control problem are provided. In particular, it is shown that under appropriate assumptions the open-loop bilinear control function is unique. The resulting optimality system is discretized by the Chang-Cooper scheme that guarantees positivity of the forward solution. The effectiveness of the proposed computational framework is validated with a stochastic Lotka-Volterra model and a noised limit cycle model.

Optimal control of probability density functions of stochastic processes

Abstract A Fokker‐Planck framework for the formulation of an optimal control strategy of stochastic processes is presented. Within this strategy, the control objectives are defined based on the probability density functions of the stochastic processes. The optimal control is obtained as the minimizer of the objective under the constraint given by the Fokker‐Planck model. Representative stochastic processes are considered with different control laws and with the purpose of attaining a final target configuration or tracking a desired trajectory. In this latter case, a receding‐horizon algorithm over a sequence of time windows is implemented.

Canonical and noncanonical equilibrium distribution

We address the problem of the dynamical foundation of noncanonical equilibrium. We consider, as a source of divergence from ordinary statistical mechanics, the breakdown of the condition of time scale separation between microscopic and macroscopic dynamics. We show that this breakdown has the effect of producing a significant deviation from the canonical prescription. We also show that, while the canonical equilibrium can be reached with no apparent dependence on dynamics, the specific form of noncanonical equilibrium is, in fact, determined by dynamics. We consider the special case where the thermal reservoir driving the system of interest to equilibrium is a generator of intermittent fluctuations. We assess the form of the noncanonical equilibrium reached by the system in this case. Using both theoretical and numerical arguments we demonstrate that Lévy statistics are the best description of the dynamics and that the Lévy distribution is the correct basin of attraction. We also show that the correct path to noncanonical equilibrium by means of strictly thermodynamic arguments has not yet been found, and that further research has to be done to establish a connection between dynamics and thermodynamics.

A Fokker–Planck Feedback Control-Constrained Approach for Modelling Crowd Motion

ABSTRACT A Fokker–Planck control approach to model crowd motion is investigated. This strategy is formulated as a bilinear optimal control-constrained problem governed by the Fokker–Planck equation modeling the evolution of the probability density function of the stochastic motion of the crowd. Theoretical results on existence and regularity of controls are provided. For computational purposes, the resulting optimality system is discretized using an alternate-direction implicit Chang–Cooper scheme that guarantees conservativeness, positivity, L2 stability, and second-order accuracy of the forward solution. A projected non-linear conjugate gradient scheme is used to solve the optimality system. Results of numerical experiments demonstrate the efficiency of the proposed control framework.

Stochastic versus dynamic approach to Lévy statistics in the presence of an external perturbation

Abstract We study the influence of a dissipation process on diffusion dynamics triggered by slow fluctuations. We study both strong- and weak-friction regimes. When the latter regime applies, the system is attracted by the basin of either Gauss or Levy statistics according to whether the fluctuation correlation function is integrable or not. We analyze with a numerical calculation the border between the two basins of attraction.

Analysis of splitting methods for solving a partial integro-differential Fokker-Planck equation

A splitting implicit-explicit (SIMEX) scheme for solving a partial integro-differential Fokker-Planck equation related to a jump-diffusion process is investigated. This scheme combines the Chang-Cooper method for spatial discretization with the Strang-Marchuk splitting and first- and second-order time discretization methods. It is proved that the SIMEX scheme is second-order accurate, positive preserving, and conservative. Results of numerical experiments that validate the theoretical results are presented.

Analysis Of Upwind Method For Piecewise Deterministic Markov Processes

Abstract A numerical upscaling approach, NU, for solving multiscale elliptic problems is discussed. The main components of this NU are: i) local solve of aux-iliary problems in grid blocks and formal upscaling of the obtained results to build a coarse scale equation; ii) global solve of the upscaled coarse scale equation; and iii) reconstruction of a fine scale solution by solving local block problems on a dual coarse grid. By its structure NU is similar to other methods for solving multiscale elliptic problems, such as the multiscale finite element method, the multiscale mixed finite element method, the numerical subgrid upscaling method, heterogeneous mul-tiscale method, and the multiscale finite volume method. The difference with those methods is in the way the coarse scale equation is build and solved, and in the way the fine scale solution is reconstructed. Essential components of the presented here NU approach are the formal homogenization in the coarse blocks and the usage of so called multipoint flux approximation method, MPFA. Unlike the usual usage of MPFA as a discretization method for single scale elliptic problems with tensor discontinuous coefficients, we consider its usage as a part of a numerical upscaling approach. An aim of this paper is to compare the performance of NU with the one of MsFEM for ceratin multiscale problems. In particular, it is shown that the resonance effect, which limits the application of the Multiscale FEM, does not appear, or it is significantly relaxed, when the presented here numerical upscaling approach is applied.

FOKKER–PLANCK-BASED CONTROL OF A TWO-LEVEL OPEN QUANTUM SYSTEM

The control of a two-level open quantum system subject to dissipation due to environment interaction is considered. The evolution of this system is governed by a Lindblad master equation which is augmented by a stochastic term to model the effect of time-continuous measurements. In order to control this stochastic master equation model, a Fokker–Planck control framework is investigated. Within this strategy, the control objectives are defined based on the probability density functions of the two-level stochastic process and the controls are computed as minimizers of these objectives subject to the constraints represented by the Fokker–Planck equation. This minimization problem is characterized by an optimality system including the Fokker–Planck equation and its adjoint. This optimality system is approximated by a second-order accurate, stable, conservative and positive-preserving discretization scheme. The implementation of the resulting open-loop controls is realized with a receding-horizon algorithm over a sequence of time windows. Results of numerical experiments demonstrate the effectiveness of the proposed approach.

On the Action of a Semi-Markov Process on a System of Differential Equations

Abstract We deal with a model equation for stochastic processes that results from the action of a semi-Markov process on a system of ordinary differential equations. The resulting stochastic process is deterministic in pieces, with random changes of the motion at random time epochs. By using classical methods of probability calculus, we first build and discuss the fundamental equation for the statistical analysis, i.e. a Liouville Master Equation for the distribution functions, that is a system of hyperbolic PDE with non-local boundary conditions. Then, as the main contribute to this paper, by using the characteristics’ method we recast it to a system of Volterra integral equations with space fluxes, and prove existence and uniqueness of the solution. A numerical experiment for a case of practical application is performed.