Existence of a Potential for Dissipative Dynamical Systems

Graham, R.; Tél, Tamás

doi:10.1103/physrevlett.52.9

Cited by 217 publications

(215 citation statements)

References 12 publications

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“…Thus, all the extrema of the stationary Lagrangian are determined to be stable minima of the time-dependent dynamic system. Note however, that it is unlikely that a true potential exists over all M G space [122,157].…”

Section: General Descriptionmentioning

confidence: 99%

Statistical mechanics of neocortical interactions: A scaling paradigm applied to electroencephalography

Ingber

1991

Phys. Rev. A

124

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A series of papers has developed a statistical mechanics of neocortical interactions (SMNI), deriving aggregate behavior of experimentally observed columns of neurons from statistical electrical-chemical properties of synaptic interactions. While not useful to yield insights at the single neuron level, SMNI has demonstrated its capability in describing large-scale properties of short-term memory and electroencephalographic (EEG) systematics. The necessity of including nonlinear and stochastic structures in this development has been stressed. In this paper, a more stringent test is placed on SMNI: The algebraic and numerical algorithms previously developed in this and similar systems are brought to bear to fit large sets of EEG and evoked potential data being collected to investigate genetic predispositions to alcoholism and to extract brain "signatures" of short-term memory. Using the numerical algorithm of Very Fast Simulated Re-Annealing, it is demonstrated that SMNI can indeed fit this data within experimentally observed ranges of its underlying neuronal-synaptic parameters, and use the quantitative modeling results to examine physical neocortical mechanisms to discriminate between high-risk and low-risk populations genetically predisposed to alcoholism. Since this first study is a control to span relatively long time epochs, similar to earlier attempts to establish such correlations, this discrimination is inconclusive because of other neuronal activity which can mask such effects. However, the SMNI model is shown to be consistent with EEG data during selective attention tasks and with neocortical mechanisms describing short-term memory previously published using this approach. This paper explicitly identifies similar nonlinear stochastic mechanisms of interaction at the microscopic-neuronal, mesoscopic-columnar and macroscopic-regional scales of neocortical interactions. These results give strong quantitative support for an accurate intuitive picture, portraying neocortical interactions as having common algebraic or physics mechanisms that scale across quite disparate spatial scales and functional or behavioral phenomena, i.e., describing interactions among neurons, columns of neurons, and regional masses of neurons.PA

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Section: General Descriptionmentioning

confidence: 99%

Statistical mechanics of neocortical interactions: A scaling paradigm applied to electroencephalography

Ingber

1991

Phys. Rev. A

124

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show abstract

“…Graham and co-workers [5,6] have argued that a potential can exist with equation (17) if the above Hamiltonian is integrable for H = 0, because the condition implies that there exist a smooth separatrix which connect smoothly the stable and unstable manifolds emanating from the hyperbolic fixed points of the dynamical system.…”

Section: Existence Of a Potential For Dissipative Semiclassical Symentioning

confidence: 99%

“…In a significant analysis Graham and coworkers [5,6] had examined the condition for existence of potential for classical dissipative systems. We now extend this analysis to the present semiclassical context.…”

Section: Existence Of a Potential For Dissipative Semiclassical Symentioning

confidence: 99%

“…These studies have enriched our understanding in several theoretical issues, e. g. , the symmetry between the growth and the decay of classical fluctuations in equilibrium and its breakdown under nonequilibrium conditions [3] , the existence of a nonequilibrium potential of a dissipative system [5,6] etc.…”

Section: Introductionmentioning

confidence: 99%

“…The essential description of the physical situation rests on the Fokker-Planck equations for the probability distribution functions of the relevant variables of the dynamics. In the weak noise limit the fluctuations have been described [3][4][5][6] by appropriate auxiliary Hamiltonian or path integral methods.…”

Section: Introductionmentioning

confidence: 99%

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