Neural Computations by Networks of Oscillators

We describe here how a network of oscillators can perform neural computations. In particular, it shown how the connectivity within the network can be created to memorize data in terms of phase relations between synchronized states. The memorized states are extracted through correlation calculations. The influence of noise on the system is discussed. 1 Electronic Circuit Models of Neural Activity An important aspect of modeling neural activity has been the development of oscillatory electronic circuit analogs. Early in this was Lapicque (1907) who introduced a low pass filter with a circuit-breaker: Charge accumulates on a capacitor until a threshold voltage is reached, then a circuit breaker closes and the charge is flushed. Around 1920, van der Pol and others developed multiple plate vacuum tube circuits that exhibited self-sustained oscillation. Variable resistance shunting circuits were described by Hodgkin and Huxley in their work in 1952, and updates of van der Pol’s circuit involving tunnel diodes were studied by FitzHugh and others around 1960. Networks were studied using filter banks by Longuet-Higgins and Gabor and by Wilson and Cowan around 1970. In 1980, Hoppensteadt introduced phase locked loop (PLL) models to study phase-locking phenomena observed in neural tissue. These developments are described in [l, 21. A particularly interesting aspect of PLLs is that they are quite stable in the presence of noise. A typical PLL model is described by a voltage wave form, say V , whose argument (its phase) is 4(t). So, the output is described by V(4(t)). A major breakthrough made by engineers was to use frequency domain methods to model complex circuits. In particular, a PLL is described by the system: T i + z = P(4 , t ) 4 = w + z where i = dz /d t , etc., z is the output of a low-pass filter with time constant T, w is the center frequency of a voltage controlled oscillator, and P is an oscillatory function o f t and 4 that describes the output of a phase detector. A useful approximation in many applications is to remove the filter (i.e., set T = 0), and the resulting model is 0 = w + P(4, t ) It is easy to connect such circuits together to make networks like those considered here. 41 0-7695-06 19-4/00 $10.00