A frequency domain technique for characterizing nonlinearities in biological systems

doi:10.1016/S0022-5193(88)80323-0

Journal of Theoretical Biology

Volume 133, Issue 3, 7 August 1988, Pages 293-317

https://doi.org/10.1016/S0022-5193(88)80323-0 Get rights and content

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 F₁ and F₂ Hz, the output of a rectifier contains multiple discrete frequency components of frequency (nF₁ ± mF₂) 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 (nF₁ ± mF₂) 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 F₁ Hz flicker of fixed modulation depth superimposed on F₂ 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 ΔF = ΔT⁻¹, where ΔT 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.

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^*: Parts of this research were supported by the Canadian NSERG and MRC and the NEI, and sponsored by the U.S. Air Force Office of Scientific Research.

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A frequency domain technique for characterizing nonlinearities in biological systems*

Vision Res.

Vision Res.

Vision Res.

Vision. Res.

Vision Res.

Brain Res.

Brain Res.

Biophys. J.

J. Mag. Mag. Materials

J. Physiol.

J. Physiol.

Bell System Tech. J.

J. Physiol.

Auditory cortex

Ann. N. Y. Acad. Sci.

J. Physiol.

J. Physiol., Lond.

Vision Res.

J. Physiol.

J. Inst. elect. Eng., Lond.

J. acoust. Soc. Am.

On the Theory of Functions of a Real Variable and the Theory of Fourier's Series

J. Physiol.

On the Direct Numerical Calculation of Elliptic Functions and Integrals

J. opt. Soc. Am.

Analysis of Physiological Systems. The White Noise Approach