1. Experimental studies have shown that a central pattern generator in the spinal cord of the lamprey can produce the basic rhythm for locomotion. This pattern generator interacts with the reticular neurons forming a spinoreticulospinal loop. To better understand and investigate the mechanisms for locomotor pattern generation in the lamprey, we examine the dynamic behavior of a simplified neural network model representing a unit spinal pattern generator (uPG) and its interaction with the reticular system. We use the techniques of bifurcation analysis and specifically examine the effects on the dynamic behavior of the system of 1) changing tonic drives to the different neurons of the uPG; 2) altering inhibitory and excitatory interconnection strengths among the uPG neurons; and 3) feedforward-feedback interactions between the uPG and the reticular neurons. 2. The model analyzed is a qualitative left-right symmetric network based on proposed functional architecture with one class of phasic reticular neurons and three classes of uPG neurons: excitatory (E), lateral (L), and crossed (C) interneurons. In the model each class is represented by one left and one right neuron. Each neuron has basic passive properties akin to biophysical neurons and receives tonic synaptic drive and weighted synaptic input from other connecting neurons. The neuron's output as a function of voltage is given by a nonlinear function with a strict threshold and saturation. 3. With an appropriate set of parameter values, the voltage of each neuron can oscillate periodically with phase relationships among the different neurons that are qualitatively similar to those observed experimentally. The uPG alone can also oscillate, as observed experimentally in isolated lamprey spinal cords. Varying the parameters can, however, profoundly change the state of the system via different kinds of bifurcations. Change in a single parameter can move the system from nonoscillatory to oscillatory states via different kinds of bifurcations. For some parameter values the system can also exhibit multistable behavior (e.g., an oscillatory state and a nonoscillatory state). The analysis also shows us how the amplitudes of the oscillations vary and the periods of limit cycles change as different bifurcation points are approached. 4. Altering tonic drive to just one class of uPG neurons (without altering the interconnections) can change the state of the system by altering the stability of fixed points, converting fixed points to oscillations, single oscillations to two stable oscillations, etc. Two-parameter bifurcation diagrams show the critical regions in which a balance between the tonic drives is necessary to maintain stable oscillations. A minimum tonic drive is necessary to obtain stable oscillatory output. With appropriate changes in the tonic drives to the L and C neurons, stable oscillatory output can be obtained even after eliminating the E neurons. Indeed, the presence of active E neurons in the biological system does not prove they play a functional role in the system, because tonic drive from other sources can substitute for them. On the other hand, very high excitation of any one class of neurons can terminate oscillations. Appropriate balance of tonic drives to different neuron classes can help sustain stable oscillations for larger tonic drives. Published experimental results concerning changes in amplitude and swimming frequency with increased tonic drives are mimicked by the model's responses to increased tonic drive. 5. Interconnectivity among the neurons plays a crucial role. The analysis indicates that the C and L classes of neurons are essential components of the model network. Sufficient inhibition from the L to C neurons as well as mutual inhibition between the left and right halves is necessary to obtain stable oscillatory output. When the E neurons are present in the model network, they must receive appropriate tonic drive and provide appropriate excitation
Dynamic Behavior of the Solutions for a Two-layer Neural Network Model with Delays
