Optimization of olfactory model in software to give 1/f power spectra reveals numerical instabilities in solutions governed by aperiodic (chaotic) attractors.
We present a general connectionist model for an olfactory system. The dynamical behavior of each node (neural ensemble) of the model is governed by a second-order ordinary differential equation (ODE) followed by an asymmetric sigmoidal function, relating the aggregate activity of neurons to system parameters and stimuli from an outside environment. A digital implementation of the general connectionist model simulates the characteristics of a mammalian olfactory system having modifiable synaptic connections and spatio-temporal interactions among neural ensembles. Each of four distributed delay terms is represented by a second-order ODE. A parameter optimization algorithm is an integral component of the model. The parameter optimization discussed in this paper results in aperiodic oscillations having a near 1/f-type power spectrum with a peak in the gamma range, simulating the electro-encephalographic (EEG) potentials from the neural olfactory system. Random optimization is used for a rough search in a global parameter domain and the parameter self-adaptation rule serves for fine tuning within a local domain after the global search. By design the model is started from an unstable zero point by an external impulse input. It requires 650-850ms to pass through an initializing transient before settling into a strange attractor. The attractor persists for at least 1500ms, which is 7.5-20 times longer than the duration of the maximal stationary states observed in the EEGs and it is stable under perturbation by simulated sensory inputs giving response amplitudes less than 2 times the basal aperiodic activity. However, the optimization to 1/f-type activity reveals an inherent limit on digital simulation of chaotic states, owing to attractor crowding such that the size of basins decreases with increasing size of the model, until it approaches the size of digitizing step in computation, here a 64-bit word ( approximately 10(-16)). Outputs that are not optimized to approach 1/f-type power spectra transit earlier to limit cycle activity, usually well before 2000ms. The duration of stationarity is increased by randomizing the terminal bit of the 64-bit words representing the state variables. The 1/f-type solutions are also exquisitely sensitive to parameter truncation; parameter values must be saved in their full binary form for re-starting. The implications in terms of numerical instability, chaos, attractor crowding and the shadowing theorem are discussed.
Burke
Brian C.
BC
Division of Neurobiology, LSA 129 and Department of Molecular and Cell Biology, University of California, Berkeley, USA
Freeman
Walter J.
WJ
Chang
Hung Jen
HJ