Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 84 / 3

At rest, the cell has relatively high intracellular potassium and high extracellular sodium. The membrane potential is negative inside, largely because selective resting permeability, particularly potassium leak, combines with these electrochemical gradients. The sodium-potassium pump maintains the gradients over time; it is not the immediate regenerative current of an action potential.

Depolarization rapidly activates sodium voltage-gated ion channels, causing inward sodium current and further depolarization. Slower sodium inactivation and delayed potassium activation then reduce inward current and increase outward potassium current, repolarizing the membrane. Persisting potassium conductance can produce an after-hyperpolarization; recovery of sodium availability contributes to the neuronal refractory period. A spike transfers only a small fraction of the bulk ions, so it changes charge separation much more than it changes the overall concentration gradients.

For two-state ion-channel gating, let $X$ be the fraction of permissive gates. Opening transfers probability from $1-X$ at rate ${\alpha }_X(V)$, and closing transfers it back at rate ${\beta }_X(V)$:Independent subunits that must all be permissive multiply their probabilities. The Hodgkin-Huxley model therefore uses $g_{\mathrm{Na}}={\bar{g}}_{\mathrm{Na}}{m}^{3}h$ and $g_{\mathrm{K}}={\bar{g}}_{\mathrm{K}}{n}^4$: $m$ describes sodium activation, $h$ sodium availability, and $n$ potassium activation. Their voltage-dependent time scales produce the sequence of inward and outward current. This channel-level construction is the mechanism in the primary Hodgkin–Huxley membrane-current model.

A dynamic clamp is a real-time feedback experiment. Measure the membrane potential through an electrode, numerically evolve chosen channel or synapse state variables, calculate the modeled current, and inject that current into the real neuron. A virtual synapse might use $I_{\mathrm{syn}}=g_{\mathrm{syn}}(t)(E_{\mathrm{syn}}-V(t))$; a voltage-dependent virtual channel also updates its gate probabilities from the measured voltage. Repeat at a rate fast relative to the dynamics being modeled. Unlike voltage clamp, this procedure does not hold the voltage at a command value.

Two uses are to add a controlled inhibitory conductance and measure its effect on firing or integration, and to couple two real neurons by artificial reciprocal inhibitory synapses to test the resulting network dynamics. Both artificial conductances and reciprocal inhibitory connections were demonstrated in the primary Sharp–O'Neil–Abbott–Marder dynamic-clamp experiments. Conductance magnitude, time course and reversal voltage can be varied without requiring the cell to express a new channel. Sampling delay, electrode error and the inability of a single injection site to reproduce all dendritic conductance locations are practical limitations.

The leaky integrate-and-fire model treats the membrane as a capacitance $C_m$. The leak conductance $g_L$ drives voltage towards its resting reversal $V_L$; the excitatory conductance $g_E$ drives it towards its reversal $V_E$; and $I_e$ is injected current, positive inward. A sufficiently high $V_E$ makes increasing $g_E$ depolarizing over the relevant voltage range, while also increasing total conductance and shortening the integration time.

For constant input and conductances, putThenIf this trajectory reaches $V_{\theta }$, register a spike and reset the voltage to $V_0<V_{\theta }$, optionally enforcing a specified neuronal refractory period. The reset makes repeated threshold crossings possible under a constant suprathreshold drive. If $V_{\infty }\le V_{\theta }$ and the initial voltage is below threshold, no finite-time crossing occurs; equality gives asymptotic approach. For time-varying currents or conductances, integrate the subthreshold differential equation until the next crossing.

Use this model when spike timing, spike counts and network integration matter more than the detailed waveform: it is inexpensive for large recurrent networks or comparisons of current-driven and conductance-driven input. It omits ionic spike initiation, channel-specific adaptation and many dendritic effects, for which a Hodgkin-Huxley model or a spatial model may be needed.

Assume constant current, $g_L>0$, $C_m>0$, and reset below threshold. With $g_E=0$, $V_{\infty }=V_L+I_e/g_L$. A spike occurs in finite time after reset precisely when $V_{\infty }>V_{\theta }$, so the critical boundary isAt equality the voltage approaches threshold asymptotically, so the finite-time firing condition is the strict inequality $I_e>I_c$.

Solve the exponential voltage trajectory from reset to threshold. The interspike interval of this constant-input firing rate of a leaky integrate-and-fire neuron isThusThe logarithm decreases strictly with current, so $r$ increases, starting from zero as $I_e\downarrow I_c$. At large current, its argument increment is small andThe second expression states the leading linear growth, not an exact intercept formula. If an explicit refractory time $t_{\mathrm{ref}}$ is added, the rate instead becomes $1/(T+t_{\mathrm{ref}})$ and approaches $1/t_{\mathrm{ref}}$ at very high current. The printed model contains no such extra interval, which is why its unrestricted high-current rate is linear.