### 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 language | English (US) |
---|---|

Pages (from-to) | 175-181 |

Number of pages | 7 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 148 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 1 2001 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Physica D: Nonlinear Phenomena*,

*148*(1-2), 175-181. https://doi.org/10.1016/S0167-2789(00)00189-5

}

*Physica D: Nonlinear Phenomena*, vol. 148, no. 1-2, pp. 175-181. https://doi.org/10.1016/S0167-2789(00)00189-5

**Differentiability implies continuity in neuronal dynamics.** / Francis, Joseph T.; So, Paul; Gluckman, Bruce; Schiff, Steven.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Differentiability implies continuity in neuronal dynamics

AU - Francis, Joseph T.

AU - So, Paul

AU - Gluckman, Bruce

AU - Schiff, Steven

PY - 2001/1/1

Y1 - 2001/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0040333700&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040333700&partnerID=8YFLogxK

U2 - 10.1016/S0167-2789(00)00189-5

DO - 10.1016/S0167-2789(00)00189-5

M3 - Article

VL - 148

SP - 175

EP - 181

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-2

ER -