A Fundamental Inequality Governing the Rate Coding Response of Sensory Neurons

Abstract

A fundamental inequality governing the spike activity of peripheral neurons is derived and tested against auditory data. This inequality states that the steady-state firing rate must lie between the arithmetic and geometric means of the spontaneous and peak activities during adaptation. Implications towards the development of auditory mechanistic models are explored.

Appendices

Appendix A: Computer code to solve for adaptation response

The following code was developed for the MATLAB programming environment (MathWorks, version R2023a) but can be easily adapted to any other language to solve (1)–(4) numerically with only a few lines of code. The parameters can be set to any positive value. A forward Euler method is used to solve the differential equation in (4).

While the above code was developed to solve for the adaptation response, a simple change to the line governing the stimulus allows for the code to solve the response to any time-varying input. For the example illustrated in the code, note that $\text {SR}=0.35$, ${\text {PR}}=1.15$ and ${\text {SS}}=0.75$ satisfy the upper bound of the inequality (13).

Appendix B: Slope of steady-state activity at low intensities

There is an alternative way to express (13): for low intensities, the slope of steady-state activity SS with respect to $\bar{k}$ equals one-half the slope of peak activity PR. We will now prove this assertion. Consider differentiable functions f(x), g(x) and h(x) defined over the domain $x \ge 0$ satisfying the inequality $f(x) \le g(x) \le h(x)$ for all non-negative values of x. Moreover, impose the conditions $f(0)=g(0)=h(0)=y_0$ and $f'(0)=h'(0)$. $f'$ designates the first derivative of f with respect to x, etc. Clearly these statements are equivalent to $g'(0)=f'(0)=h'(0)$ by the definition of the derivative.

Next we recast the problem in terms of SS, PR and SR. In this case, intensity replaces x and $\text {GM}(0)={\text {SS}}(0)=\text {AM}(0)=\text {SR}$ where AM and GM refer to the arithmetic and geometric means of SR and PR. That is, at zero intensity both the GM and AM equals SR. Moreover since SR is constant with respect to intensity, both $\text {AM}'(0)$ and $\text {GM}'(0)$ equals ${\text {PR}}'(0)/2$. By the same steps as before, we conclude that ${\text {SS}}'(0)=\text {AM}'(0)=\text {GM}'(0)={\text {PR}}'(0)/2$. Therefore, the slope of SS equals one-half the value of PR at zero intensity, i.e. $\bar{k} =\bar{k}_{\text {sp}}$.