Neuronal Dynamics (2)
Ion Channels and the Hodgkin-Huxley Model
The online version of this chapter:
Chapter 2 Ion Channels and the Hodgkin-Huxley Model https://neuronaldynamics.epfl.ch/online/Ch2.html
In the previous description, spikes are formal events that are generated at the moment of threshold crossing.
Now we will talk about how a spike can be generated based on the biophysics of cell membranes.
2.1 Equilibrium potential
2.1.1 Nernst potential
The probability of a molecule to take a state of energy \(E\) is proportional to the Boltzman factor \(p(t) \propto \exp (-E/kT)\) where \(k\) is the Boltzman constant and \(T\) the temperature.
The probality to find an ion in the region around location \(x\) proportional to \(\exp [-qu(x)/kT]\).
Write \(n(x)\) for the ion density at point \(x\). \[ \frac{n(x_1)}{n(x_2)}=\exp [-\frac{qu(x_1)-qu(x_2)}{kT}] \tag{2.1} \]
Q: What's the meaning of "Since this is a statement about an equilibrium state, the reverse must also be true"?
Nernst potential (能斯特势) \[ \Delta u=\frac{kT}{q}\ln \frac{n_2}{n_1} \tag{2.2} \]
2.1.2 Reversal Potential
The cell membrane bilayer lipids - a nearly perfect electrical insulator specific proteins as ion gates - ion pumps: actively transport ions from one side to the other. - ion channels: passively
Resting Potential The value \(u_{rest}\) is determined by the dynamic equilibrium between the ion flow through the channels (permeability of the membrane) and active ion transport (efficiency of the ion pump in maintaining the concentration difference).
2.2 Hodgkin-Huxley Model
Four equations \[ \begin{aligned} C &\frac{\mathrm{d}u}{\mathrm{d}t} =-g_Kn^{4}(u-E_K)-g_{Na}m^{3}h(u-E_{Na})-g_l(u-E_l)+I(t) \\ &\frac{\mathrm{d}}{\mathrm{d}t}n =-\frac{n-n_0(u)}{\tau_n(u)} \\ &\frac{\mathrm{d}}{\mathrm{d}t}m =-\frac{m-m_0(u)}{\tau_m(u)} \\ &\frac{\mathrm{d}}{\mathrm{d}t}h =-\frac{h-h_0(u)}{\tau_h(u)} \end{aligned} \]
Gating variables \[ I_{ion}=-g_{ion}r^{n_1}s^{n_2} \] \(r\) represents activation variables, \(r_0(u)\to 1, u\to \infty\). \(s\) represents inactivation variables, \(s_0(u)\to 0, u\to \infty\) > There could be 2 gating variables for some ions. > It turns out to be useful to distinguish between a deactivated channel (\(m\) close to zero and \(h\) close to one) and an inactivated channel ( \(h\) close to zero)
Threshold in the Hodgkin Huxley Model
Here, threshold relates to the paradigm of the current.
threshold for repetitive firing current threshold (constant current)
threshold current: \(\theta_i\). If we are above this minimal current, the model generates regular firing. If not, then the model shows constant firing. threshold for action potential current threshold (step current) (uninformative)
pulse current Consider \(I(t)=q\delta(t-t_0)\). It causes a jump of the membrane potential.
Fixing duration of the current, we have a current threshhold. If the current is above the threshhold, then the neuron gives an action potential.
SAP: spike after potential
step current input \((I_1,\Delta I,I_2)\)
The final current as well as the step size that come into account.
ramp input
Type II/Class II Behavior(Hodgking-Huxley model with standard parameters, giant axon of squid): Using a very slow ramp input on a Hodgkin-Huxley model, we find an f-I curve with a jump
Type I/Class I Behavior(Hodgking-Huxley model with other parameters, e.g. for cortical pyramidal neuron): Using the frame variable of Hodgking-Huxley models but change parameters so it's more adapted to cortical neurons, then you make it a smooth response with very low frequencies.
Stochastic Channel Opening
Channels open stochastically.
Example: Time Constants, Transition Rates, and Channel Kinetics The activation and inactivation dynamics of each channel type can be described in terms of voltage-dependent transition rates \(\alpha\) and \(\beta\), \[ \frac{\mathrm{d}m}{\mathrm{d}t}=\alpha_m(u)(1-m)-\beta_m(u)m \] \[ \frac{\mathrm{d}n}{\mathrm{d}t}=\alpha_n(u)(1-n)-\beta_n(u)n \] \[ \frac{\mathrm{d}h}{\mathrm{d}t}=\alpha_h(u)(1-h)-\beta_h(u)h \] The asymptotic value \(x_0(u)\) and the time constant \(\tau_x(u)\) are given by \(x_0(u)=\alpha_x(u)/[\alpha_x(u)+\beta_x(u)]\) and \(\tau_x(u)=[\alpha_x(u)+\beta_x(u)]^{-1}\).
\(n\) can be intepreted as the probability of finding a single potassium channel open. In a patch with \(K\) channels, approximately \(k \thickapprox (1-n)K\) channels are expected to be closed. We may interpret \(\alpha_n(u)\Delta t\) as the probability that in a short time interval \(\Delta t\) one of the momentarily closed channels switches to the open state.
2.3 The Zoo of Ion Channels
Framework for biophysical neuron models
Sodium Ion Channels and the Type-I Regime
Adaptation and Refractoriness
I have a constant input current, an adaptation means that in the spike intervals, get longer and longer.
Ex: \[ I_{M}= g_{M} m(u-E_{k}) \] - Potassium current - Kv7 subunits - slow time constant
A current such as \(I_{M}\) is one of the potentially many sources of adaptation. It works by lowering the membrane potential, by lowering the spike after potential(SAP).
Another way of generating adaptation: not by changing SAP, but by increasing the firing threshold.
\[ I_{NaP} = g_{NaP} mh(u-E_{Na}) \] - persistent sodium current - fast activation time constant - slow inactivation (~1s)