The role of the cerebellum in motor control and perception.
- M G Paulin
- Brain, behavior and evolution
The spiking of a vestibular sensory neuron can be interpreted as an assertion about the kinematic state of the head. By defining firing times as fuzzy sets, these assertions can be defuzzified to give a standard firing time orfiring rate code. Firing time andfiring rate are special cases of a rich language in which neural spikes can be interpreted using fuzzy membership functions. The Pacini-Kosko wacking filter, applied as a model of neural control of eye stabilization during head movements, illustrates how fuzzy rules may be applied to fuzzy data to produce crisp behaviour, and how the behaviour of individual neurons in the brain may be understood using fuzzy theory. Sensory neurons from the vestibular apparatus of the inner ear tell the brain about head movements. Afferents from the horizontal semicircular canals fire when the head rotates about its vertical axis. In conventional models of vestibular sensory coding, head angular velocity is encoded in the firing rate of these neurons and a neuron's behaviour is characterised using a transfer function or an equivalent dynamical model r.11. However, individual neurons operate asynchronously, and because of the uncertainty principle [Z] the instantaneous firing rate of a neuron is undefined. Thus, while conventional models of vestibular coding deal with neural firing rates at particular times, neither rute nor time can be crisply defined in this system. I have previously examined the issue of interpreting neural firing patterns using conventional signal processing theory , showing how neural firing rate estimates can be optimized within the constraints of the uncertainty principle and the Shannon sampling theorem. Simultaneous uncertainty in pulse timing and pulse rate is minimised if the pulses are modelled as gaussian wave packets extending through time, rather than as discrete events occurring at points in time. This is an unrealistic picture but it does give the right answer. That is, it minimises error in translating back and forth between pulse timing and pulse rate representations of neural firing . Fuzzy set theory provides an intuitively realistic route to the same end. We can regard action potentials as discrete events in fuzzy time. In fuzzy time, rime t is a fuzzy set rather than a single point. By analogy with the result of the conventional analysis , a natural fuzzification is to define the membership function of the fuzzy set time t to be a unit height gaussian curve centred at crisp time t. An event occurring at crisp time t+T occurs at fuzzy time t with degree e ~ p ( ? ~ / a ~ ) . The scale parameter a quantifies the vagueness of assertions about time. At one extreme (a+O), fuzzy now means crisp now, at the other extreme (a+-) fuzzy now means any time, and in general fuzzy now means about now. The code can be defuzzified to give a timing model because the mean of each fuzzy firing time is a crisp firing time, and it can be defuzzified to give a rate model because firing rate at crisp time t is 1/(&0) times the sum of the degrees of membership of crisp time t in all fuzzy firing times. In the timing case, fuzzy time t defuzzifies to crisp time t. In the rate case, the factor U(* a) is required if we want to interpret the result in units of spikes per time unit. This is because the fuzzy membership functions are normalised so that crisp time t is fuzzy time t with degree 1, but in the rate code each spike must make a total contribution of 1 (unit of power) to the rate. Formally, defuzzifying the fuzzy code to give a rate code is a trivial reinterpretation of conventional filtering for rate estimation where a determines the bandwidth of the rate estimating filter. The optimality argument carries over from the conventional case and it implies that this fuzzy time model gives the most accurate possible representation of an inherently illdefined situation . 1 This fuzzy model is equivalent to a conventional model. That is, in the fuzzy framework we may use the Same procedure as in the conventional framework to obtain a firing time or firing rate model of neural coding, but we have a different interpretation of what