Intrinsic oscillations in neural networks: A linear model for the nth-order loop

Abstract A theoretical map for the characteristic modes of neurolectric oscillation in an arbitrary number ( n ) of nerve cells arranged in a single closed loop is presented. A linear model with time lag is used. Generally, there are n possible damped (or rising) oscillatory states for every 2π range of frequency, in addition to possible nonoscillating solutions. Usually, higher frequencies of oscillation correspond to faster decay rates among characteristic states. Increasing the number of cells in the loop increases the number of allowable characteristic states; increasing the (geometric) mean interconnection coefficient increases the endurance of the leading characteristic state; and increasing the average interunit conduction time increases the number of characteristic states that endure relatively long as compared to representative times for the system. Various endurance criteria are mapped on the parameter space of the network. In qualitative terms, all states tend to die out quickly in rapidly conducting and weakly interconnected loops; all states but one die out quickly in rapidly conducting and strongly interconnected loops; many states may endure for several membrane time constants in slowly conducting and weakly connected loops; and many modes may endure for considerably longer than representative times in slowly conducting and strongly interconnected loops. A tentative linking of the properties of a loop's oscillations to the physical characteristics of its nerve cells is presented. It is suggested that the so-called characteristic states of a given subsystem might be envisioned collectively as a repertoire of intrinsic dynamic modes available to the system, and could provide a means whereby the properties of the entire network interrelate to achieve its subjective or behavioral function.