Minimal time spiking in various ChR2-controlled neuron models

Abstract

We use conductance based neuron models, and the mathematical modeling of optogenetics to define controlled neuron models and we address the minimal time control of these affine systems for the first spike from equilibrium. We apply tools of geometric optimal control theory to study singular extremals, and we implement a direct method to compute optimal controls. When the system is too large to theoretically investigate the existence of singular optimal controls, we observe numerically the optimal bang–bang controls.

Appendices

Appendix 1: Numerical constants for the Morris–Lecar model

The numerical values of the several constants and their physiological meaning are taken from Ditlevsen and Greenwood (2013) and gathered in Table 5.

Table 5 Meaning and numerical values of the constants appearing in the Morris–Lecar model

Table 6 gathers the numerical values for Fig. 10.

Table 6 Meaning and numerical values of the constants, taken from Saint-Hilaire and Longtin (2004), appearing in the Morris–Lecar model

Appendix 2: Numerical constants for the Hodgkin–Huxley model

$$\begin{aligned} \alpha _n(V)&= \frac{0.1-0.01V}{e^{1-0.1V}-1},&\beta _n(V)&= 0.125e^{-\frac{V}{80}},\\ \alpha _m(V)&= \frac{2.5-0.1V}{e^{2.5-0.1V}-1},&\beta _m(V)&= 4e^{-\frac{V}{18}},\\ \alpha _h(V)&= 0.07e^{-\frac{V}{20}},&\beta _h(V)&= \frac{1}{e^{3-0.1V}+1}. \end{aligned}$$

The following Table 7 gathers the numerical values of the Hodgkin–Huxley model, as given in the original paper Hodgkin and Huxley (1952).

Table 7 Meaning and numerical values of the constants appearing in the Hodgkin–Huxley model

The equilibrium potential $E_L$ of the leakage current is usually set so that the equilibrium value of the (HH) system is such that $V=0$.

Appendix 3: Numerical constants for the ChR2 models

1.1 The 3-states model

The constants of the model are the rates $K_d$ and $K_r$ of the transitions between the open state and the light adapted closed state and between the two closed states, the maximal conductance $g_{ChR2}$ and the equilibrium potential $V_{ChR2}$. As specified in Sect. 3, we assume that these rates are constants during the evolution in order to obtain an affine control system. For the numerical computations, we took the values given in Table 1 of Nikolic et al. (2009):

$$\begin{aligned} K_d = 0.2\text { m\,s}^{-1}, \quad K_r = 0.021 \text { m\,s}^{-1}. \end{aligned}$$

The maximal conductance is given by the formula $g_{ChR2} = \rho _{ChR2}g^*_{ChR2}$, with $\rho _{ChR2}$ the density of channels and $g^*_{ChR2}$ the conductance of a single channel. These values are taken from Foutz et al. (2012) to obtain

$$\begin{aligned} g_{ChR2} = 0.65 \text { mS}\, \text {cm}^{-2}. \end{aligned}$$

As mentioned at the end of “Appendix 2”, the physiological equilibrium membrane potential is mathematically shifted to equal 0. The equilibrium potential of the ChR2 that is usually measured around 0 (Foutz et al. 2012) and very often taken as 0 (Foutz et al. 2012; Nikolic et al. 2009). The exact value 0 would raise a mathematical problem because since we shifted the value of $E_L$ so that $V=0$ corresponds to the equilibrium point of the uncontrolled system we start from. Indeed, $V=0$ would also correspond to an equilibrium point of the controlled system, regardless of the value of the control. For this reason, we shifted the value of $V_{ChR2}$ and took it equal to 60mV. This value corresponds to the shift of the membrane resting potential for the Morris–Lecar and Hodgkin–Huxley models.

Finally we can give an estimation of the physiological maximal value $u_{max}$ of the control. Indeed, upon illumination, the transition rate between the dark adapted closed state and the open state in Nikolic et al. (2009) is $\varepsilon F$ where $\varepsilon =0.5$ is the quantum efficiency and F is given by the formula

$$\begin{aligned} F = \frac{\sigma _{ret} \phi }{w_{loss}}, \end{aligned}$$

where $\sigma _{ret} \simeq 10^{-8} \upmu \text {m}^2$ is the retinal cross section (cross section of the photon receptor on the ChR2), $\phi = 6.2\times 10^{9}\text { ph}\cdot \upmu \text {m}^{-2}\,\text {s}^{-1}$ is the original flux of photons and $w_{loss} =1.1$ is a loss factor. As for the numerical value of $K_d$ and $K_r$ we took the one of Table 1 in Nikolic et al. (2009) for the value of $\phi$. With these values we get

$$\begin{aligned} u_{max} = 0.028 \text { m\,s}^{-1}. \end{aligned}$$

1.2 The 4-states model

The numerical values for the ChR2-4-states model are taken from Foutz et al. (2012) and gathered in Table 8.

Table 8 Numerical values of the constants appearing in the ChR2-4-States model