Stochastic neurodynamics. (English) Zbl 0686.92009
Summary: Stochastic dynamics of relative membrane potential in the neural network is investigated. It is called stochastic neurodynamics. The least action principle for stochastic neurodynamics is assumed, and used to derive the fundamental equation. It is called a neural wave equation. A solution of the neural wave equation is called a neural wave function and describes stochastic neurodynamics completely. Linear superposition of neural wave functions provides us with a mathematical model of associative memory process.
As a simple application of stochastic neurodynamics, a mathematical representation of static neurodynamics in terms of equilibrium statistical mechanics of spin systems is derived.
As a simple application of stochastic neurodynamics, a mathematical representation of static neurodynamics in terms of equilibrium statistical mechanics of spin systems is derived.
MSC:
| 92Cxx | Physiological, cellular and medical topics |
| 60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |
| 82B99 | Equilibrium statistical mechanics |
Keywords:
neural holography; Stochastic dynamics of relative membrane potential; neural network; stochastic neurodynamics; least action principle; neural wave equation; neural wave function; Linear superposition; associative memory process; static neurodynamics; equilibrium statistical mechanics of spin systemsReferences:
| [1] | Abeles, M. (1982). Local Cortical Circuits, Springer-Verlag, New York. · doi:10.1007/978-3-642-81708-3 |
| [2] | Albeverio, S.; Blanchard, Ph.; Høegh-Krohn, R., Newtonian diffusions and planets, with a remark on non-standard Dirichlet forms and polymers, 1-24 (1984), Berlin, Heidelberg · Zbl 0547.60082 |
| [3] | Albeverio, S., Blanchard, Ph., Høegh-Krohn, R. and Mebkhout, M. (1986). Strata and voids in galactic structures—a probabilistic approach, BiBoS (preprint). |
| [4] | Blanchard, Ph., Combe, Ph. and Zheng, W. (1987). Mathematical and Physical Aspects of Stochastic Mechanics, Lecture Notes in Physics 281, Springer-Verlag, New York. · Zbl 0628.60104 |
| [5] | Buhmann, J. and Schulten, K. (1986). Influence of noise on the behavior of an autoassociative neural network, AIP Conf. Proc., 151, 71-76. · Zbl 1356.82009 · doi:10.1063/1.36222 |
| [6] | Hopfield, J. (1982). Neural network and physical systems with emergent collective computational abilities, Proc. Nat. Acad. Sci. U.S.A., 79, 2554-2558. · Zbl 1369.92007 · doi:10.1073/pnas.79.8.2554 |
| [7] | Hopfield, J. and Tank, D. (1985). Neural computation of decisions in optimization problems, Biol. Cybernet., 52, 141-152. · Zbl 0572.68041 |
| [8] | Kato, T. (1966). Perturbation Theory for Linear Operators, Springer-Verlag, New York. · Zbl 0148.12601 |
| [9] | Nagasawa, M. (1980). Segregation of a population in an environment. J. Math. Biol., 9, 213-235. · Zbl 0447.92017 · doi:10.1007/BF00276026 |
| [10] | Nagasawa, M. (1981). An application of the segregation model for septation of escherichia coli, J. Theoret. Biol., 90, 445-455. · doi:10.1016/0022-5193(81)90298-8 |
| [11] | Nelson, E. (1967). Dynamical Theories of Brownian Motions, Princeton University Press, Princeton. · Zbl 0165.58502 |
| [12] | Nelson, E. (1984). Quantum Fluctuations, Princeton University Press, Princeton. · Zbl 0563.60001 |
| [13] | Pribram, K. (1971). The Languages of the Brain, Prentice Hall Englewood Cliffs, New Jersey. |
| [14] | Yasue, K. (1981a). Quantum mechanics and stochastic control theory, J. Math. Phys., 22, 1010-1020. · doi:10.1063/1.525006 |
| [15] | Yasue, K. (1981b). Stochastic calculus of variations, J. Funct. Anal., 41, 327-340. · Zbl 0482.60063 · doi:10.1016/0022-1236(81)90079-3 |
| [16] | Yasue, K. (1983). A variational principle for the Navier-Stokes equation, J. Funct. Anal., 51, 133-141. · Zbl 0533.35075 · doi:10.1016/0022-1236(83)90021-6 |
| [17] | Zambrini, J.-C. (1985). Stochastic dynamics: A review of stochastic calculus of variations, Internat. J. Theoret. Phys., 24, 277-327. · Zbl 0563.60098 · doi:10.1007/BF00669792 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
