Abstract

Recent work has identified nonlinear deterministic structure in neuronal dynamics using periodic orbit theory. Troublesome in this work were the significant periods of time where no periodic orbits were extracted - "dynamically dark" regions. Tests for periodic orbit structure typically require that the underlying dynamics are differentiable. Since continuity of a mathematical function is a necessary but insufficient condition for differentiability, regions of observed differentiability should be fully contained within regions of continuity. We here verify that this fundamental mathematical principle is reflected in observations from mammalian neuronal activity. First, we introduce a null Jacobian transformation to verify the observation of differentiable dynamics when periodic orbits are extracted. Second, we show that a less restrictive test for deterministic structure requiring only continuity demonstrates widespread nonlinear deterministic structure only partially appreciated with previous approaches.

Original languageEnglish (US)
Pages (from-to)175-181
Number of pages7
JournalPhysica D: Nonlinear Phenomena
Volume148
Issue number1-2
DOIs
StatePublished - Jan 1 2001

Fingerprint

Differentiability
continuity
Periodic Orbits
Orbits
orbits
Imply
Differentiable
Verify
Period of time
Null
Necessary
Demonstrate
Observation

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

Cite this

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title = "Differentiability implies continuity in neuronal dynamics",
abstract = "Recent work has identified nonlinear deterministic structure in neuronal dynamics using periodic orbit theory. Troublesome in this work were the significant periods of time where no periodic orbits were extracted - {"}dynamically dark{"} regions. Tests for periodic orbit structure typically require that the underlying dynamics are differentiable. Since continuity of a mathematical function is a necessary but insufficient condition for differentiability, regions of observed differentiability should be fully contained within regions of continuity. We here verify that this fundamental mathematical principle is reflected in observations from mammalian neuronal activity. First, we introduce a null Jacobian transformation to verify the observation of differentiable dynamics when periodic orbits are extracted. Second, we show that a less restrictive test for deterministic structure requiring only continuity demonstrates widespread nonlinear deterministic structure only partially appreciated with previous approaches.",
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Differentiability implies continuity in neuronal dynamics. / Francis, Joseph T.; So, Paul; Gluckman, Bruce; Schiff, Steven.

In: Physica D: Nonlinear Phenomena, Vol. 148, No. 1-2, 01.01.2001, p. 175-181.

Research output: Contribution to journalArticle

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