The olfactory system is constantly solving pattern-recognition problems by the creation of a large space to codify odour representations and optimizing their distribution within it. A model of the Olfactory Bulb was developed by Z. Li and J. J. Hopfield Li and Hopfield (1989) based on anatomy and electrophysiology. They used nonlinear simulations observing that the collective behavior produce an oscillatory frequency. Here, we show that the Subthreshold hopf bifurcation is a good candidate for modeling the bulb and the Subthreshold subcritical hopf bifurcation is a good candidate for modeling the olfactory cortex. Network topology analysis of the subcritical regime is presented as a proof of the importance of synapse plasticity for memory functions in the olfactory cortex.
Abstract This article describes computer simulation of the dynamics of a distributed model of the olfactory system that is aimed at understanding the role of chaos in biological pattern recognition. The model is governed by coupled nonlinear differential equations with many variables and parameters, which allow multiple high-dimensional chaotic states. An appropriate set of the parameters is identified by computer experiments with the guidance of biological measurements, through which this model of the olfactory system maintains a low dimensional global chaotic attractor with multiple “wings.” The central part of the attractor is its basal chaotic activity, which simulates the electroencephalographic (EEG) activity of the olfactory system under zero signal input (exhalation). It provides the system with a ready state so that it is unnecessary for the system to “wake up” from or return to a “dormant” equilibrium state every time that an input is given (by inhalation). Each of the wings may be either a near-limit cycle (a narrow band chaos) or a broad band chaos. The reproducible spatial pattern of each near-limit cycle is determined by a template made in the system. A novel input with no template activates the system to either a nonreproducible near-limit cycle wing or a broad band chaotic wing. Pattern recognition in the system may be considered as the transition from one wing to another, as demonstrated by the computer simulation. The time series of the manifestations of the attractor are EEG-like waveforms with fractal dimensions that reflect which wing the system is placed in by input or lack of input. The computer simulation also shows that the adaptive behavior of the system is scaling invariant, and it is independent of the initial conditions at the transition from one wing to another. These properties enable the system to classify an uninterrupted sequence of stimuli. The olfactory bulb of mammals aids in the discrimination of odors. A mathematical model based on the bulbar anatomy and electrophysiology is described. Simulations of the highly non-linear model produce a 35---60 Hz modulated activity which is coherent across the bulb. The decision states (for the odor information) in this system can be thought of as stable cycles, rather than point stable states typical of simpler neuro-computing models. Analysis shows that a group of coupled non-linear oscillators are responsible for the oscillatory activities. The output oscillation pattern of the bulb is determined by the odor input. The model provides a framework in which to understand the transform between odor input and the bulbar output to olfactory cortex. There is significant correspondence between the model behavior and observed electrophysiology.