Nonlinear behavior in small neural systems

This work addresses the nonlinear behavior of one or two model neurons under the influence of different stimuli, whether they be forms of chaos control or varieties of added noise. This is a step towards the ultimate objective of exploring the notion that a neural system might utilize a mechanism such as a memory-searching chaotic attractor to locate and retrieve stable-memory limit cycles.; The biological realism of the Hopfield neuron models is discussed, and the concept of an “effective” neuron is introduced. The dynamical effects of adding inertial/inductance terms to an effective-neuron system are presented along with arguments for the biological relevance of such terms. A two neuron system with one or two inertial terms added is shown to exhibit chaos. The chaos is confirmed by Lyapunov exponents, power spectra, and phase-space plots.; The effects of multiplicative and additive noise on the dynamics of a two effective-neuron system are investigated. One of the neurons possesses an added inertial term so the system is able to generate chaotic dynamics. The multiplicative noise is added to the connection parameter J ₁₁, and the additive noise is added to the equation for ₂ like an external driving force. Using J₁₁ as a bifurcation parameter, the system is examined as it passes from limit cycle dynamics to chaotic dynamics. Both types of noise are found to lower the bifurcation point with respect to its deterministic value, and both cause the dynamics to expand in phase space. For equivalent levels of noise, additive noise is found to have a stronger effect on the dynamics than multiplicative noise. The bifurcation points are explored by means of ensembles of the largest Lyapunov exponents derived from the stochastic dynamics.; A brief overview is presented of the current state of control theory in chaotic systems. One control method, Hübler's [74] technique of using aperiodic forces to drive nonlinear oscillators to resonance, is analyzed. The technique is applied to a single effective-neuron with an added inertial term, and it is verified through analysis of the power spectrum, force, resonance, and energy transfer of the system.; The entrainment control method put forth by Jackson and Grosu [82] is also explored. A modification to this technique is examined, where Gaussian white noise, representing the natural noise inherent in any real dynamical system, is added to the control mechanism, making it stochastic. The stochastic entrainment control is applied to an effective-neuron system as a way to extract stable limit cycles, or memories, from a chaotic attractor, which represents a memory-searching state. The higher the noise level applied to the system, the more convergent phase-space trajectories become, as reflected in the average values of the largest Lyapunov exponents. This suggests that the noise inherent in a system might not inhibit the retrieval of memories, as one might first suspect. The average values of the correlation dimensions increase with added noise, which could signal an increase in the overall complexity of the system.