A frequency domain technique for characterizing nonlinearities in biological systems
Abstract
Response asymmetry (e.g. to light ON vs. OFF), a frequently encountered property of neurons at both peripheral and central levels in sensory pathways, can be modelled as a rectifier. When fed with the sum of two sinusoids of frequencies F 1 and F 2 Hz, the output of a rectifier contains multiple discrete frequency components of frequency (nF 1 ± mF 2) where n and m are zero or integers. We describe a double Fourier series method for obtaining the amplitudes and phases of these components for physiologically relevant neural models including (1) compressive, linear, accelerating and mixed compressive/accelerating single model neurons, (2) a cascaded series of model neurons and (3) the parallel/cascaded case corresponding to dichoptic or dichotic stimulation. If one of the inputs is held constant while the other's amplitude is varied, we obtain a family of curves-one for each (nF 1 ± mF 2) component. The family of curves seems to be characteristic of the particular non-linear system.
Neural models may be tested by first calculating the family of curves and then comparing these theoretical data with a physiologically measured family of curves. Illustrating this approach, we compare the predictions of case (1) and case (2) models with the family of curves obtained by stimulating with F 1 Hz flicker of fixed modulation depth superimposed on F 2 Hz flicker of variable modulation depth. Evoked brain responses were analysed by non-destructive zoom-FFT giving resolution up to the Heisenberg-Gabor limit of Δ F = Δ T -1, where Δ T is the recording duration. Thus, for example, the spectrum could contain 50 000 lines over a D.C.-100 Hz bandwidth for a 500 second recording duration. Empirically, the bandwidth of discrete physiological components is no more than 0·004 Hz. Like the Wiener kernel time-domain approach, this frequency-domain approach is restricted to time-invariant systems with a limited settling time and without appreciable hysteresis. On the other hand, the Weiner kernel approach seems to handle dynamic systems more conveniently. But this frequency-domain method has the important advantage that different signal components are well separated from each other and from noise, whereas signal and noise overlap in the time domain. Consequently, high-order terms are more easily recorded using the frequency domain method, thus giving sharper distinctions between candidate non-linear models.- Publication:
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Journal of Theoretical Biology
- Pub Date:
- 1988
- DOI:
- 10.1016/S0022-5193(88)80323-0
- Bibcode:
- 1988JThBi.133..293R