### Abstract

Is the characterization of biological systems as complex systems in the mathematical sense a fruitful assertion? In this paper we argue in the affirmative, although obviously we do not attempt to confront all the issues raised by this question. We use the fly's visual system as an example and analyse our experimental results of one particular neuron in the fly's visual system from this point of view. We find that the motion-sensitive 'H1' neuron, which converts incoming signals into a sequence of identical pulses or 'spikes', encodes the information contained in the stimulus into an alphabet composed of a few letters. This encoding occurs on multilayered sets, one of the features attributed to complex systems. The conversion of intervals between consecutive occurrences of spikes into an alphabet requires us to construct a generating partition. This entails a one-to-one correspondence between sequences of spike intervals and words written in the alphabet. The alphabet dynamics is multifractal both with and without stimulus, though the multifractality increases with the stimulus entropy. This is in sharp contrast to models generating independent spike intervals, such as models using Poisson statistics, whose dynamics is monofractal. We embed the support of the probability measure, which describes the distribution of words written in this alphabet, in a two-dimensional space, whose topology can be reproduced by an M-shaped map. This map has positive Lyapunov exponents, indicating a chaotic-like encoding.

Original language | English |
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Pages (from-to) | 345-357 |

Number of pages | 13 |

Journal | Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences |

Volume | 366 |

Issue number | 1864 |

Early online date | 2 Aug 2007 |

DOIs | |

Publication status | Published - 13 Feb 2008 |

### Keywords

- complex systems
- neuron dynamics
- multifractality
- dynamical systems
- generating partitions
- strange attractors
- symbolic encodings
- template analysis
- periodic-orbits
- dimensions

## Cite this

*Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences*,

*366*(1864), 345-357. https://doi.org/10.1098/rsta.2007.2093