Biological Cybernetics

, Volume 33, Issue 4, pp 237-247

First online:

Nonlinear gain mediating cortical stimulus-response relations

  • Walter J. FreemanAffiliated withDepartment of Physiology-Anatomy, University of California

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Single isolated neurons show a nonlinear increase in likelihood of firing in response to random input, when they are biased toward threshold by steady-state depolarization. It is postulated that this property holds for neurons in the olfactory bulb, a specialized form of cortex in which the steady state is under centrifugal control. A model for this non-linearity is based on two first order differential equations that interrelate three state variables: activity density in the pulse mode p of a local population (subset) of neurons; activity density in the wave mode u (mean dendritic current); and an intervening variable m that may be thought to represent the average subthreshold value in the subset for the sodium activation factor as defined in the Hodgkin-Huxley equations. A stable steady state condition is posited at (p 0, u 0, m 0). It is assumed that: (a) the nonlinearity is static; (b) m increases exponentially with u; (c) p approaches a maximum p m asymptotically as m increases; (d) p≧0; and (e) the steady state values of p 0, u 0, and m 0 are linearly proportional to each other. The model is tested and evaluated with data from unit and EEG recording in the olfactory bulb of anesthetized and waking animals. It has the following properties. (a) There is a small-signal near-linear input-output range. (b) There is bilateral saturation for large input, i.e. the gain defined as dp/du approaches zero for large |u|. (c) The asymptotes for large input±u are asymmetric. (d) The range of output is variable depending on u 0. (e) Most importantly, the maximal gain occurs for u>u 0, so that positive (excitatory) input increases the output and also the gain in a nonlinear manner. It is concluded that large numbers of neurons in the olfactory bulb of the waking animal are maintained in a sensitive nonlinear state, that corresponds to the domain of the subthreshold local response of single axons, as it is defined by Rushton and Hodgkin.