On the rate coding response of peripheral sensory neurons

Abstract

The rate coding response of a single peripheral sensory neuron in the asymptotic, near-equilibrium limit can be derived using information theory, asymptotic Bayesian statistics and a theory of complex systems. Almost no biological knowledge is required. The theoretical expression shows good agreement with spike-frequency adaptation data across different sensory modalities and animal species. The approach permits the discovery of a new neurophysiological equation and shares similarities with statistical physics.

Appendices

A The optimal sample size

The optimal sample size $m_\mathrm{eq}$ is the number of samples after which m no longer changes. Since the fluctuation scaling law posits that the variance of a signal increases with the magnitude of the signal, a choice of constant $m_\mathrm{eq}$ implies that the estimation error in the mean will increase when intensity is increased. On the other hand, if the standard error $\sqrt{\sigma ^2/m_\mathrm{eq}}$ is held constant, $m_\mathrm{eq}$ must then take the form

$$\begin{aligned} m_\mathrm{eq} \propto \left( I+\delta I\right) ^p \end{aligned}$$

(36)

requiring sample size to change significantly with intensity.

Between these two extremes lies a third possibility. Consider the situation where the sensory system is presented with an input of intensity $I_1$ which is later changed to $I_2$. Without loss of generality, assume that $I_2>I_1$. At steady-state, the standard error of $I_1$ is $\text {SE}_1$. Let the initial uncertainty in $I_2$ be $\text {SE}_\text {2,initial}$. Increasing the number of samples will cause this error to fall. How is the steady-state error in $I_2$ determined? Taking $\text {SE}_\text {2}$ to be the geometric average of the standard errors, we obtain

$$\begin{aligned} \text {SE}_2=\left( \text {SE}_1 \times \text {SE}_{\text {2,initial}} \right) ^{1/2} \end{aligned}$$

(37)

That is, the error in estimating $I_2$ is equal to the average of the steady-state error in $I_1$ and the initial uncertainty in $I_2$.

To see what effect (37) has on the optimal sample size, let $m_2(0)$ be the initial sample size just after the change in intensity. Since m must remain continuous across the boundary we have $m_2(0)=m_\mathrm{eq,1}$, where $m_\mathrm{eq,1}$ is the optimal sample size for $I_1$. This calculation assumes that steady-state is achieved prior to the change in intensity. From this, we conclude that the expression $\sigma ^2/m_\mathrm{eq}^2$ is invariant to changes in intensity. Thus, the general relationship between optimal sample size and intensity is

$$\begin{aligned} m_\mathrm{eq} =c (I+\delta I)^{p/2} \end{aligned}$$

(38)

where c is a constant. For simplicity, c is set to unity as it can be incorporated into $\beta$ in (10). Equation (38) is the expression for optimal sample size used in the theory and can be tested both directly and indirectly through comparison with experimental data.

B Key assumptions of the theory

• 1. The sensory receptor draws repeated, independent samples of the stimulus magnitude to estimate the mean.

• 2. Samples are processed with limited resolution resulting in normally distributed error with zero mean and variance R, which is constant relative to the input.

• 3. Firing rate is proportional to the Boltzmann–Shannon measure of uncertainty in the mean.

• 4. The statistics of the sensory signal are governed by a Tweedie distribution.

• 5. The signal mean $\mu$ is a sum of the experimenter controlled intensity I plus constant additive background noise $\delta I$.

• 6. Sampling rate is a function of the difference between the current and optimal sample sizes.

• 7. The optimal sample size is determined from an average of standard errors in the mean.