Subthreshold dynamics of a single neuron from a Hamiltonian perspective

We use Hamilton’s equations of classical mechanics to investigate the behavior of a cortical neuron on the approach to an action potential. We use a two-component dynamic model of a single neuron, due to Wilson, with added noise inputs. We derive a Lagrangian for the system, from which we construct Hamilton’s equations. The conjugate momenta are found to be linear combinations of the noise input to the system. We use this approach to consider theoretically and computationally the most likely manner in which such a modeled neuron approaches a firing event. We find that the firing of a neuron is a result of a drop in inhibition, due to a temporary increase in negative bias of the mean noise input to the inhibitory control equation. Moreover, we demonstrate through theory and simulation that, on average, the bias in the noise increases in an exponential manner on the approach to an action potential. In the Hamiltonian description, an action potential can therefore be considered a result of the exponential growth of the conjugate momenta variables pulling the system away from its equilibrium state, into a nonlinear regime.

Figure 1

A plot of the full “fast” solution to Hamilton’s equations for the Wilson neuron for the case of $I_{\mathrm{dc}}=21.475\mu \mathrm{A}{\mathrm{cm}}^{-2}$, ${\sigma }_1=0.02\mu \mathrm{A}{\mathrm{cm}}^{-2}{\mathrm{ms}}^{1∕2}$, and ${\sigma }_2=0.02{\mathrm{ms}}^{1∕2}$. Parts (a)–(d) show the variables $q_1=V$, $q_2=R$, $p_1$, and $p_2$, respectively, as functions of time. Part (e) shows the growth in $-p_2$ at small times on a log scale; part (f) shows the growth closer to the firing event. Note how the two time scales are very different, corresponding to the slow (e) and fast (f) scales.

Figure 2

A plot of the pseudokinetic (dash) and pseudopotential (dot-dash) energies as a function of time, for the simulation of Fig. 1. The sum of the two (solid line) is conserved.

Figure 3

A comparison of the simulated Wilson neuron and the Hamiltonian approach, for the case of $I_{\mathrm{dc}}=21.475\mu \mathrm{A}{\mathrm{cm}}^{-2}$, ${\sigma }_1=0.02\mu \mathrm{A}{\mathrm{cm}}^{-2}{\mathrm{ms}}^{1∕2}$, and ${\sigma }_2=0.02{\mathrm{ms}}^{1∕2}$. The Hamiltonian solution using the “slow” approximation is shown by the thick line; the mean result of the 119 acceptable simulations out of 10 000 trials is shown by the thinner line. The spread in the simulated trajectories is indicated by the dotted lines which are placed one standard deviation above and below the mean trajectory. Part (a) shows the solution for the membrane potential $V$; part (b) shows the solution for the recovery variable $R$. The sequencies have been aligned so that they cross $-55\mathrm{mV}$ at $t=0\mathrm{ms}$, therefore the time sequences are not all exactly $43.2\mathrm{ms}$ long.

Figure 4

A plot of the pseudokinetic (dash) and pseudopotential (dot-dash) energies as a function of the time before a spike, for the Hamiltonian simulation of Fig. 3 with a slow approximation. The total pseudoenergy (solid line) is conserved at early times but not in the final $5\mathrm{ms}$ before the firing event.

Figure 5

A comparison of the simulated Wilson neuron and the slow Hamiltonian approach, for the case of $I_{\mathrm{dc}}=15.0\mu \mathrm{A}{\mathrm{cm}}^{-2}$, ${\sigma }_1=0.1\mu \mathrm{A}{\mathrm{cm}}^{-2}{\mathrm{ms}}^{1∕2}$, and ${\sigma }_2=0.1{\mathrm{ms}}^{1∕2}$. There were 80 sequences accepted from 36 000 trials. The Hamiltonian solution is shown by the thick line; the mean result of the simulations by the thinner line. The spread in the simulated trajectories is indicated by the dotted lines which are placed one standard deviation above and below the mean trajectory. Part (a) shows the solution for the membrane potential $V$; part (b) shows the solution for the recovery variable $R$.

Figure 6

A comparison of the simulated Wilson neuron and the slow Hamiltonian approach, for the case of $I_{\mathrm{dc}}=21.475\mu \mathrm{A}{\mathrm{cm}}^{-2}$, ${\sigma }_1=0.02\mu \mathrm{A}{\mathrm{cm}}^{-2}{\mathrm{ms}}^{1∕2}$, and ${\sigma }_2=0.02{\mathrm{ms}}^{1∕2}$. The Hamiltonian solution is shown by the thick gray line; the mean result of the 75 accepted simulations out of 10 000 attempts is shown by the thinner solid line. The spread in the simulated trajectories is indicated by the dashed lines which are placed one standard deviation above and below the mean trajectory. Also shown by the dot-dash lines are the standard uncertainty in the mean (i.e., the standard deviation divided by $\sqrt{75-1}$, where 75 is the number of accepted simulations); these lines are one standard uncertainty above and below the mean, respectively. Part (a) shows the solution for the membrane potential $V$; part (b) shows the solution for the recovery variable $R$.

Figure 7

A plot of the mean noise input to the Wilson model over the $15\mathrm{ms}$ preceding a spike, for the case of $I_{\mathrm{dc}}=21.4\mu \mathrm{A}{\mathrm{cm}}^{-2}$, ${\sigma }_1=0.03\mu \mathrm{A}{\mathrm{cm}}^{-2}{\mathrm{ms}}^{1∕2}$, and ${\sigma }_2=0.03{\mathrm{ms}}^{1∕2}$. A total of 190 simulations were accepted from 500 trials. (a) The mean noise input ${\sigma }_1{\xi }_1$ (solid line) to the membrane potential equation (3.1) averaged over $1\mathrm{ms}$ bins before the spike. The dot-dash lines indicate the standard uncertainty in the mean. (b) The mean noise input ${\sigma }_2{\xi }_2$ to the recovery variable equation (3.2) (solid line) and its standard uncertainty (dot-dash lines). (c) The same data as (b), plotted on a logarithmic scale. The growth is well fitted by an exponential (straight line).

Figure 8

A plot of the noise input to the Wilson model over the $50\mathrm{ms}$ preceding a spike, for the case of $I_{\mathrm{dc}}=21.4\mu \mathrm{A}{\mathrm{cm}}^{-2}$ and the lower noise input of ${\sigma }_1=0.015\mu \mathrm{A}{\mathrm{cm}}^{-2}{\mathrm{ms}}^{1∕2}$ and ${\sigma }_2=0.015{\mathrm{ms}}^{1∕2}$. A total of 98 simulations were accepted from 500 attempts. (a) The mean simulated noise input ${\sigma }_1{\xi }_1$ (solid line) to the membrane potential equation (3.1) averaged over $3\mathrm{ms}$ bins before the spike. The dot-dash lines indicate the standard uncertainty in the mean. (b) The mean simulated noise input ${\sigma }_2{\xi }_2$ to the recovery variable equation (3.2) (solid line) and its standard uncertainty (dot-dash lines). Note how the reduction in noise input to the recovery variable occurs over a long time-frame before the spike. (c) The same data as (b), plotted on a logarithmic scale. An exponential fit (straight line) has been applied.

Figure 9

(Color online) A plot of the mean trajectories for the approach to a firing event, for the case of $I_{\mathrm{dc}}=21.475\mu \mathrm{A}{\mathrm{cm}}^{-2}$, ${\sigma }_1=0.02\mu \mathrm{A}{\mathrm{cm}}^{-2}{\mathrm{ms}}^{1∕2}$, and ${\sigma }_2=0.02{\mathrm{ms}}^{1∕2}$. A total of 7759 out of 40 000 simulations were accepted. (a) The mean membrane potential, grouped by time taken to fire. The dashed line indicate the standard uncertainties in the mean for one trajectory; the other uncertainties are lower than this. The groups are (in terms of time before firing occurs): $5–10\mathrm{ms}$ (black), $10–15\mathrm{ms}$ (violet), $15–20\mathrm{ms}$ (blue), $20–25\mathrm{ms}$ (green), $25–30\mathrm{ms}$ (yellow), $30–35\mathrm{ms}$ (orange), and $35–40\mathrm{ms}$ (red). Each sequence has been time aligned so they reach the same value of $V$ simultaneously. For the $5–10\mathrm{ms}$ sequence, only the last $5\mathrm{ms}$ have been plotted so that all sequences in the group contribute at every point in the graph; similarly for the other sequences. (b) The mean recovery variable, grouped in a similar way. (c) The mean random input to the recovery variable equation (3.2), over periods of $0.5\mathrm{ms}$, with simulations grouped in the same way as part (a).

Figure 10

(Color online) A plot of the mean trajectories for the approach to a firing event, in terms of the deviation from equilibrium values, for the same case as Fig. 9. (a) The natural logarithm of the mean deviation of the membrane potential from its equilibrium value, grouped by time taken to fire. The groups are (in terms of time before firing occurs): $5–10\mathrm{ms}$ (black), $10–15\mathrm{ms}$ (violet), $15–20\mathrm{ms}$ (blue), $20–25\mathrm{ms}$ (green), $25–30\mathrm{ms}$ (yellow), $30–35\mathrm{ms}$ (orange), and $35–40\mathrm{ms}$ (red). Each sequence has been time aligned so they reach the same value of $V$ simultaneously. For the $5–10\mathrm{ms}$ sequence, only the last $5\mathrm{ms}$ have been plotted so that all sequences in the group contribute at every point in the graph; similarly for the other sequences. (b) The natural logarithm of the (negative) of the deviation of the mean recovery variable from its equilibrium value, grouped in a similar way. (c) The natural logarithm of the (negative) mean random input to the recovery variable equation (3.2), over periods of $0.5\mathrm{ms}$, with simulations grouped in the same way as part (a).

Figure 11

(Color online) A plot of the mean trajectories for the approach to a firing event, in terms of the deviation from equilibrium values, for the case of $I_{\mathrm{dc}}=21.475\mu \mathrm{A}{\mathrm{cm}}^{-2}{\mathrm{ms}}^{1∕2}$, ${\sigma }_1=0.005\mu \mathrm{A}{\mathrm{cm}}^{-2}{\mathrm{ms}}^{1∕2}$, and ${\sigma }_2=0.005{\mathrm{ms}}^{1∕2}$. A total of 1509 out of 10 000 simulations were accepted. (a) The natural logarithm of the mean deviation of the membrane potential from its equilibrium value, grouped by time taken to fire. The groups are (in terms of time before firing occurs): $100–200\mathrm{ms}$ (black), $200–300\mathrm{ms}$ (blue), $300–400\mathrm{ms}$ (green), and $400–500\mathrm{ms}$ (red). Each sequence has been time-aligned so they reach the same value of $V$ simultaneously. (b) The natural logarithm of the (negative) of the deviation of the mean recovery variable from its equilibrium value, grouped in a similar way. (c) The natural logarithm of the (negative) mean random input to the recovery variable equation (3.2), with simulations grouped in the same way as part (a). Note that this plot exhibits a region of exponential growth with rate constant $0.020{\mathrm{ms}}^{-1}$, between 200 and $100\mathrm{ms}$ before the spike, in agreement with the slow eigenvalue of matrix $M$.