1.Mathematical Models and Numerical Methods of Computational Neuroscience

Unit	Neural Coding and Brain Computing Unit (Tomoki Fukai)
Teacher	Alan Func
Lecture Materials	https://github.com/oist-ncbc/skill-pill-plus/tree/master/Tutorial-1_Ordinary-Differential-Equations
Library	https://github.com/iwasakishuto/University/tree/gh-pages/3A/theme/program/OIST/tutorial1.py

Brief introduction of Differential Equations

Ordinary Differential Equations¶

• A fundamental tool in theoretical Science
• The differential equation gives the dependence of derivatives $$F\left[\frac{d^nx(t)}{dt^n},\cdots,\frac{dx(t)}{dt},C\right] = 0$$
• As we can calculate the whole path ways if we know the velocity($v=dx(t)/dt$), by knowing the slopes, one may trace the whole curve!!

Examples of Models and Theories¶

Simple Harmonic Oscillator	Heat Equation(1-D)
An oscillator that is neither driven nor damped.	How the distribution of some quantity (such as heat) evolves over time in a solid medium
$$\underset{\text{Acceleration}}{\frac{d^2x(t)}{dt^2}}+\underset{\text{Returning Force}}{\omega^2x(t)} = 0$$	$$\underset{\text{Rate of Heat}}{\frac{\partial u(x,t)}{\partial t}}+\underset{\text{Returning Force}}{\alpha\frac{\partial^2u(x,t)}{\partial x^2}} = 0$$

Implementation by Numerical Method¶

• Euler Method
• Runge-Kutta Methods

Euler Method¶

• Euler method has a relatively large error.
• There are some extensions of the Euler method(one of them are Mid-Point method.)

The mid-point method is surprisingly good!!

Runge-Kutta Methods¶

• A family of implicit and explicit iterative methods.
• More: Wikipedia

Neuronal Models

To Model a Neuron...¶

• Neurons can be excited by current injection.
• If the membrance potential(Voltage) is larger than some thresholds, an action potential will be triggered.
• Membrance potential
  • means the potential difference across the neuron membrance. → Relative
  • can be controlled by concentrations of irons($\mathrm{Na}^+,\mathrm{K}^+,\mathrm{Cl}^-$)
  • is maintained by their ion bumps, and at the resting state, it is around $65\mathrm{mV}$.
  • sometimes across the threshold, and when it occurs, the neuron fires
  • works like a capacitor with a leaky current.

Leaky Integrate-and-Fire(LIF) model¶

• In 1907, Lapicque proposed a simple model(which is just the time derivative of the law of capacitance ($Q=CV$).) for neurons: $$I(t)-\frac{V_m(t) - V_{rest}}{R_m} = C_m\frac{dV_m(t)}{dt}$$
  • $I$: Input current (=const.)
  • $V_m$: Membrance potential
  • $V_{rest}$: Resting potential of the membrance
  • $R_m$: Membrane resistance (we find it is not a perfect insulator as assumed previously)
  • $C_m$: Capacity of the membrance.
• When an input current is applied, the membrane voltage increases with time until it reaches a constant threshold $V_{th}$
• At the point, a delta function spike occurs and the voltage is reset to its resting potential($V_{rest}$).
• The LIF neuron fired more frequently as the input current increases.

※ The LIF neuron fired more frequently as the input current increases.

Hodgkin-Huxley model¶

$$ I = C_m + \frac{dV_m}{dt} + g_{\mathrm{l}}\left(V_m-V_{\mathrm{l}}\right) + g_{\mathrm{K}}n^4\left(V_m-V_{\mathrm{K}}\right) + g_{\mathrm{Na}}m^3h\left(V_m-V_{\mathrm{Na}}\right)\\ \begin{cases} \begin{aligned} \frac{dn}{dt} &= \alpha_n(V_m)(1-n) + \beta_n(V_m)n\\ \frac{dm}{dt} &= \alpha_m(V_m)(1-m) + \beta_m(V_m)m\\ \frac{dh}{dt} &= \alpha_h(V_m)(1-h) + \beta_h(V_m)h\\ \end{aligned} \end{cases} $$

• Proposed by Hodgkin and Huxley in 1952 (They received the 1963 Nobel Prize in Physiology or medicin for this work.)
• Like Leaky Integrate-and-Fire(LIF) model, the input current (const.) charges up the membrance capacitor.
  • Leaky conductance
• This model considers also ion currents invoving potassium ion ($\mathrm{K}^+$) and Sodium ion ($\mathrm{Na}^{+}$) induced by the potential difference.
  • Sodium ion: has a reversal potential depolarizing the membrance. The conductivity is voltage-gated.
  • Potassium ion: has a reversal potential hyper-polarizing the membrance. The conductivity is voltage-gated.
• The individual gates act like a first order chemical reaction with two states: $$\text{Shut}\overset{\alpha}{\underset{\beta}{\rightleftarrows}}\text{Open}$$
• Propose that
  • Each $\mathrm{K}$ channel has $4$ identical activation gates.( $\mathbb{P}_{\text{open}}=n$ )
  • Each $Na$ channel has $3$ avtivation gates( $\mathbb{P}_{\text{open}}=m$ ), and $1$ inactivation gate( $\mathbb{P}_{\text{open}}=h$ ).

How the Network can be trained

Spike-Timing-Depenent Plasticity (Classical)¶

• The change in $w_{ji}$ depends on the spike timings
• In order to perform STDP with differential equatins, an online implementation of STDP is proposed $$ \begin{aligned} \tau_+\frac{dx}{dt} &= -x + a_+(x)\sum_{t_f}\delta(t-t_f)\\ \tau_-\frac{dy}{dt} &= -y + a_-(y)\sum_{t_n}\delta(t-t_n)\\ \Longrightarrow \frac{dw}{dt} &= A_+(w)x(t)\sum_{t_n}\delta(t-t_n) - A_-(w)y(t)\sum_{t_f}\delta(t-t_f) \end{aligned} $$

※ As "Pre - Post" changes...

Then, with an appropriate firing sequence, the network can be trained!!