Neural Biophysics & Networks

Derivation: The Cable Equation from First Principles

Consider a cylindrical neurite of radius $a$ with the following distributed electrical properties per unit length:

• • $r_i$: intracellular (axial) resistance per unit length ($\Omega$/cm)
• • $r_o$: extracellular resistance per unit length ($\Omega$/cm)
• • $r_m$: membrane resistance times unit length ($\Omega \cdot$cm)
• • $c_m$: membrane capacitance per unit length (F/cm)

From Kirchhoff's current law at a node of the cable: the axial current entering a segment equals the sum of the membrane current and the axial current leaving. For a segment of length $\Delta x$:

$$-\frac{\partial i_a}{\partial x} = i_m = c_m \frac{\partial V_m}{\partial t} + \frac{V_m}{r_m}$$

where $i_a$ is the axial current and $i_m$ is the membrane current per unit length. From Ohm's law for the axial current:

$$i_a = -\frac{1}{r_i + r_o} \frac{\partial V_m}{\partial x}$$

Substituting and differentiating:

$$\frac{1}{r_i + r_o} \frac{\partial^2 V_m}{\partial x^2} = c_m \frac{\partial V_m}{\partial t} + \frac{V_m}{r_m}$$

Defining the length constant and time constant:

$$\boxed{\lambda = \sqrt{\frac{r_m}{r_i + r_o}}}, \quad \boxed{\tau_m = r_m c_m}$$

The cable equation becomes:

$$\boxed{\lambda^2 \frac{\partial^2 V_m}{\partial x^2} = \tau_m \frac{\partial V_m}{\partial t} + V_m}$$

In the steady state ($\partial V_m/\partial t = 0$), the solution for a semi-infinite cable with current injection at $x = 0$ is:

$$V_m(x) = V_0 \exp(-x/\lambda)$$

The length constant $\lambda$ is the distance at which the voltage decays to$1/e \approx 37\%$ of its value at the injection site.

Physical interpretation of cable parameters: For a cylindrical neurite of radius $a$ with specific membrane resistance$R_m$ ($\Omega \cdot$cm$^2$), specific membrane capacitance $C_m$ (F/cm$^2$), and intracellular resistivity$\rho_i$ ($\Omega \cdot$cm):

$$r_m = \frac{R_m}{2\pi a}, \quad c_m = 2\pi a C_m, \quad r_i = \frac{\rho_i}{\pi a^2}$$

Substituting (and neglecting $r_o \ll r_i$ since extracellular space is large):$$\lambda = \sqrt{\frac{a R_m}{2\rho_i}}, \quad \tau_m = R_m C_m$$Note that $\lambda \propto \sqrt{a}$: larger axons have longer length constants and therefore propagate signals farther passively. This is one reason giant axons (squid, earthworm) evolved large diameters.

Derivation: Conduction Velocity in Unmyelinated Axons

For a propagating action potential of constant shape moving at velocity $v$, we can transform to a travelling coordinate $\xi = x - vt$. Then$\partial/\partial t = -v \partial/\partial \xi$ and the cable equation with active membrane current $I_{\text{ion}}$ becomes:

$$\frac{a}{2\rho_i} \frac{d^2 V}{d\xi^2} = C_m \left(-v \frac{dV}{d\xi}\right) + I_{\text{ion}}(V)$$

Dimensional analysis gives the scaling for conduction velocity. The key balance is between axial current spread (left side, proportional to $a/\rho_i$) and capacitive charging (proportional to $C_m v$). Balancing these terms:

$$\frac{a}{2\rho_i} \sim C_m v \cdot L$$

where $L$ is the spatial extent of the rising phase. With $L \sim v \tau_{\text{rise}}$and $\tau_{\text{rise}} \sim R_m C_m$, dimensional analysis yields:

$$\boxed{v \propto \sqrt{\frac{a}{2\rho_i C_m R_m}} = \frac{\lambda}{\tau_m} \propto \sqrt{a}}$$

More precisely, the full Hodgkin-Huxley analysis gives:

$$v = \sqrt{\frac{d}{4\rho_i C_m R_m}} = \frac{1}{2}\sqrt{\frac{d}{\rho_i C_m R_m}}$$

where $d = 2a$ is the diameter. For the squid giant axon ($d = 500$ $\mu$m,$\rho_i = 30$ $\Omega\cdot$cm, $C_m = 1$ $\mu$F/cm$^2$,$R_m \approx 700$ $\Omega\cdot$cm$^2$), this gives $v \approx 20$ m/s, in good agreement with experimental measurements. The $v \propto \sqrt{d}$ scaling means that to double conduction velocity, the axon diameter must quadruple — an expensive strategy in terms of space.

Derivation: Synaptic Conductance and Postsynaptic Potentials

The postsynaptic current produced by a chemical synapse is:

$$I_{\text{syn}} = g_{\text{syn}}(t)(V_m - E_{\text{syn}})$$

where $E_{\text{syn}}$ is the synaptic reversal potential. For excitatory synapses (glutamate, AMPA receptors), $E_{\text{syn}} \approx 0$ mV; for inhibitory synapses (GABA$_A$), $E_{\text{syn}} \approx -70$ mV.

The alpha-function conductance waveform provides a good phenomenological model:

$$\boxed{g_{\text{syn}}(t) = g_{\max} \cdot \frac{t}{\tau_s} \exp\left(1 - \frac{t}{\tau_s}\right)}$$

This peaks at $t = \tau_s$ with value $g_{\max}$. The postsynaptic potential is found by inserting this into the membrane equation:

$$C_m \frac{dV_m}{dt} = -\frac{V_m - V_{\text{rest}}}{R_m} - g_{\text{syn}}(t)(V_m - E_{\text{syn}})$$

For a small EPSP where $g_{\text{syn}} \ll g_m = 1/R_m$ (linear regime), the voltage deflection is approximately:

$$\Delta V_{\text{EPSP}} \approx g_{\max} R_m (E_{\text{syn}} - V_{\text{rest}}) \cdot \frac{\tau_s}{\tau_m - \tau_s} \left[\exp(-t/\tau_m) - \exp(-t/\tau_s)\right]$$

The peak EPSP occurs at:

$$t_{\text{peak}} = \frac{\tau_m \tau_s}{\tau_m - \tau_s} \ln\frac{\tau_m}{\tau_s}$$

For typical values ($\tau_m = 20$ ms, $\tau_s = 2$ ms), the EPSP peaks at $\sim 4.6$ ms. The amplitude depends on $g_{\max} R_m$: for $g_{\max} = 1$ nS and $R_m = 100$ M$\Omega$, the peak EPSP is $\sim 6$ mV.

Derivation: Leaky Integrate-and-Fire (LIF)

The subthreshold membrane dynamics are modelled as a passive RC circuit driven by external input current $I_{\text{ext}}$:

$$\boxed{\tau_m \frac{dV}{dt} = -(V - V_{\text{rest}}) + R_m I_{\text{ext}}}$$

where $\tau_m = R_m C_m$. When $V$ reaches a threshold$V_{\text{th}}$, a spike is emitted and $V$ is reset to$V_{\text{reset}}$ for a refractory period $\tau_{\text{ref}}$.

Steady-state solution: For constant input, the membrane voltage approaches:

$$V_{\infty} = V_{\text{rest}} + R_m I_{\text{ext}}$$

Spiking occurs only if $V_{\infty} > V_{\text{th}}$, i.e., when$I_{\text{ext}} > I_{\text{th}} = (V_{\text{th}} - V_{\text{rest}})/R_m$.

Firing rate derivation: For$I_{\text{ext}} > I_{\text{th}}$, the time to reach threshold from reset is found by solving the ODE with initial condition $V(0) = V_{\text{reset}}$:

$$V(t) = V_{\infty} - (V_{\infty} - V_{\text{reset}}) \exp(-t/\tau_m)$$

Setting $V(T_{\text{ISI}}) = V_{\text{th}}$ and solving for the interspike interval $T_{\text{ISI}}$:

$$V_{\text{th}} = V_{\infty} - (V_{\infty} - V_{\text{reset}}) \exp(-T_{\text{ISI}}/\tau_m)$$

$$\exp(-T_{\text{ISI}}/\tau_m) = \frac{V_{\infty} - V_{\text{th}}}{V_{\infty} - V_{\text{reset}}}$$

$$T_{\text{ISI}} = \tau_m \ln\frac{V_{\infty} - V_{\text{reset}}}{V_{\infty} - V_{\text{th}}}$$

The firing rate $f = 1/(T_{\text{ISI}} + \tau_{\text{ref}})$:

$$\boxed{f = \frac{1}{\tau_{\text{ref}} + \tau_m \ln\left(\frac{R_m I_{\text{ext}} - V_{\text{reset}} + V_{\text{rest}}}{R_m I_{\text{ext}} - V_{\text{th}} + V_{\text{rest}}}\right)}}$$

This is the f-I curve of the LIF neuron. Key features:

• • Threshold: $f = 0$ for $I < I_{\text{th}}$ (no spiking)
• • Onset: Just above threshold, $f$ rises steeply (Type I excitability)
• • Saturation: At large $I$, $f \to 1/\tau_{\text{ref}}$ (limited by refractory period)
• • Logarithmic growth: The $\ln$ function means the f-I curve is concave (sublinear), providing gain control

Derivation: The 2D Reduction

The key observation is that the HH variables separate into two timescales:

• • Fast variables: $V$ and $m$ (activation of Na$^+$). Since $m$ is fast, we approximate $m \approx m_{\infty}(V)$.
• • Slow variables: $h$ (Na$^+$ inactivation) and $n$ (K$^+$ activation). These are approximately linearly related: $h \approx 0.89 - 1.1n$.

This reduces the system to two variables: a fast "excitation" variable $v$(representing voltage/Na$^+$ activation) and a slow "recovery" variable$w$ (representing K$^+$ activation/Na$^+$ inactivation). The FitzHugh-Nagumo (FHN) equations are:

$$\boxed{\frac{dv}{dt} = v - \frac{v^3}{3} - w + I_{\text{ext}}}$$

$$\boxed{\frac{dw}{dt} = \epsilon(v + a - bw)}$$

where $\epsilon \ll 1$ controls the timescale separation, and $a, b$ are parameters. The cubic term $v - v^3/3$ mimics the N-shaped I-V curve of the fast Na$^+$ current.

Nullclines: The key geometric structures are the nullclines:

$$v\text{-nullcline: } w = v - \frac{v^3}{3} + I_{\text{ext}} \quad (\text{cubic})$$

$$w\text{-nullcline: } w = \frac{v + a}{b} \quad (\text{line})$$

The fixed point is at the intersection of the nullclines. The system's behaviour depends on where this intersection falls on the cubic:

• • Excitable regime: Fixed point on the left branch (stable). A sufficiently large perturbation sends the trajectory around the cubic (excitation), then recovery. Subthreshold perturbations decay back.
• • Oscillatory regime: As $I_{\text{ext}}$ increases, the fixed point moves to the middle branch of the cubic (unstable). A Hopf bifurcation occurs and the system exhibits sustained oscillations (repetitive firing).
• • Resting regime: At very large $I_{\text{ext}}$, the fixed point moves to the right branch (stable but depolarised). The neuron enters depolarisation block.

Hopf bifurcation analysis: The fixed point loses stability when the Jacobian has purely imaginary eigenvalues. At the fixed point$(v^*, w^*)$:$$J = \begin{pmatrix} 1 - (v^*)^2 & -1 \\ \epsilon & -\epsilon b \end{pmatrix}$$The trace $\text{tr}(J) = 1 - (v^*)^2 - \epsilon b$ changes sign when$(v^*)^2 = 1 - \epsilon b$. Since the determinant$\det(J) = \epsilon(b(1-(v^*)^2) + 1) > 0$ near the bifurcation, this is indeed a Hopf bifurcation. The critical current $I_c$ can be found by solving the nullcline intersection at $v^* = \pm\sqrt{1-\epsilon b}$.

Brain-Computer Interfaces

BCIs record neural activity (spikes or local field potentials) and decode intended actions using integrate-and-fire or more complex models of neural coding. The f-I relationship is fundamental: motor cortex neurons encode movement parameters (direction, speed) through their firing rates. Decoding algorithms extract the intended movement by inverting this relationship across a population of neurons.

Utah arrays with 96 electrodes can record from $\sim 100$ neurons simultaneously. Using population vector algorithms based on the cosine tuning model$f_i = f_{0,i} + f_{1,i}\cos(\theta - \theta_i^{\text{pref}})$, neural prosthetics achieve control of robotic arms with multiple degrees of freedom.

Computational Neuroscience

Networks of integrate-and-fire neurons form the backbone of large-scale brain simulations. Key network phenomena include:

• • Balanced excitation-inhibition: Networks maintain a balance where $E \approx I$, operating in a fluctuation-driven regime
• • Working memory: Persistent activity through recurrent excitation, modelled as attractor states
• • Oscillations: Gamma rhythms (30–80 Hz) emerge from interneuron networks (ING model) through the interplay of inhibitory synaptic time constants and network connectivity
• • Decision making: Winner-take-all dynamics between competing neural populations