2.13. Equations and numerical methods

2.13.1. Numerical methods

First-order ordinary differential equations (ODE) can be solved using different numerical methods. The method can be declared globally in the setup() call and used in all ODEs of the network:

from ANNarchy import *

setup(method='exponential')

or specified explicitely for each ODE by specifying a flag:

equations = """
    tau * dV/dt  + V =  A : init = 0.0, exponential
"""

If nothing is specified, the explicit Euler method will be used.

Different numerical methods are available:

  • Explicit Euler 'explicit'
  • Implicit Euler 'implicit'
  • Exponential Euler 'exponential'
  • Midpoint 'midpoint'
  • Event-driven 'event-driven'

Each method has advantages/drawbacks in term of numerical error, stability and computational cost.

To describe these methods, we will take the example of a system of two linear first-order ODEs:

\[ \begin{align}\begin{aligned}\frac{dx(t)}{dt} = f(x(t), y(t)) = a_x \cdot x(t) + b_x \cdot y(t) + c_x\\\frac{dy(t)}{dt} = g(x(t), y(t)) = a_y \cdot x(t) + b_y \cdot y(t) + c_y\end{aligned}\end{align} \]

The objective of a numerical method is to approximate the value of \(x\) and \(y\) at time \(t+h\) based on its value at time \(t\), where \(h\) is the discretization time step (noted dt in ANNarchy):

\[ \begin{align}\begin{aligned}x(t + h) = F(x(t), y(t))\\y(t + h) = G(x(t), y(t))\end{aligned}\end{align} \]

At each step of the simulation, the new values for the variables are computed using this update rule and will be used for the following step.

The derivative of each variable is usually approximated by:

\[ \begin{align}\begin{aligned}\frac{dx(t)}{dt} = \frac{x(t+h) - x(t)}{h}\\\frac{dy(t)}{dt} = \frac{y(t+h) - y(t)}{h}\end{aligned}\end{align} \]

The different numerical methods mostly differ in the time at which the functions \(f\) and \(g\) are evaluated.

2.13.1.1. Explicit Euler method

The explicit (forward) Euler method computes the next value for the variables by estimating their derivative at time \(t\):

\[ \begin{align}\begin{aligned}\frac{dx(t)}{dt} = \frac{x(t+h) - x(t)}{h} = f(x(t), y(t))\\\frac{dy(t)}{dt} = \frac{y(t+h) - y(t)}{h} = g(x(t), y(t))\end{aligned}\end{align} \]

so the solution is straightforward to obtain:

\[ \begin{align}\begin{aligned}x(t+h) = x(t) + h \cdot f(x(t), y(t))\\y(t+h) = y(t) + h \cdot g(x(t), y(t))\end{aligned}\end{align} \]

2.13.1.2. Implicit Euler method

The explicit (forward) Euler method computes the next value for the variables by estimating their derivative at time \(t + h\):

\[ \begin{align}\begin{aligned}\frac{dx(t)}{dt} = \frac{x(t+h) - x(t)}{h} = f(x(t+h), y(t+h))\\\frac{dy(t)}{dt} = \frac{y(t+h) - y(t)}{h} = g(x(t+h), y(t+h))\end{aligned}\end{align} \]

This leads to a system of equations which must be solved in order to find the update rule. With the linear equations defined above, we need to solve:

\[ \begin{align}\begin{aligned}\frac{x(t+h) - x(t)}{h} = a_x \cdot x(t + h) + b_x \cdot y(t + h) + c_x\\\frac{y(t+h) - y(t)}{h} = a_y \cdot x(t + h) + b_y \cdot y(t + h) + c_y\end{aligned}\end{align} \]

what gives something like:

\[ \begin{align}\begin{aligned}x(t+h) = x(t) - h \cdot \frac{ \left(a_{x} x(t) + b_{x} y(t) + c_{x} + h \left(- a_{x} b_{y} x(t) + a_{y} b_{x} x(t) + b_{x} c_{y} - b_{y} c_{x}\right)\right)}{h^{2} \left(- a_{x} b_{y} + a_{y} b_{x}\right) + h \left(a_{x} + b_{y}\right) - 1}\\y(t+h) = y(t) -h \cdot \frac{ a_{y} \left(c_{x} h + x(t)\right) + y(t) \left(- a_{y} b_{x} h^{2} + \left(a_{x} h - 1\right) \left(b_{y} h - 1\right)\right) + \left(a_{x} h - 1\right) \left(c_{y} h + y(t)\right)}{a_{y} b_{x} h^{2} - \left(a_{x} h - 1\right) \left(b_{y} h - 1\right)}\end{aligned}\end{align} \]

ANNarchy relies on Sympy to solve and simplify this system of equations and generate the update rule.

Note: This method is obviously much more computationally expensive than the explicit Euler method, although more stable. The midpoint method is a better trade-off between complexity and stability than the implicit Euler method.

2.13.1.3. Exponential Euler

The exponential Euler method is particularly stable for single first-order linear equations, of the type:

\[\tau(t) \cdot \frac{dx(t)}{dt} + x(t) = A(t)\]

The update rule is then given by:

\[x(t+h) = x(t) + (1 - \exp(- \frac{h}{\tau(t)}) ) \cdot (A(t) - x(t))\]

The difference with the explicit Euler method is the step size, which is an exponential function of the ratio \(\frac{\tau}{h}\). The accurary of the exponential Euler method on linear first-order ODEs is close to perfect, compared to the other Euler methods. As it is an explicit method, systems of equations are solved very easily with the same rule.

When the exponential method is used, ANNarchy first tries to reduce the ODE to its canonical form above (with the time constant being possibly dependent on time or inputs) and then generates the update rule accordingly.

For example, the description:

tau * dv/dt = (E - v) + g_exc * (Ee - v) + g_inh * (v - Ei)

would be first transformed in:

(1 + g_exc - g_inh) * dv/dt + v = (E + g_exc * Ee - g_inh * Ei) / (1 + g_exc - g_inh)

before being transformed into an update rule, with \(\tau(t) = 1 + g_\text{exc} - g_\text{inh}\):

\[v(t+h) = v(t) + (1 - \exp(- \frac{h}{1 + g_\text{exc} - g_\text{inh}}) ) \cdot (\frac{E + g_\text{exc} \cdot E_e - g_\text{inh} \cdot E_i}{1 + g_\text{exc} - g_\text{inh}} - v(t))\]

The exponential method can only be applied to first-order linear ODEs. Any other form of ODE will be rejected by the parser.

Important note: The step size \(1 - \exp(- \frac{h}{\tau(t)})\) is computationally expensive because of the exponential function. If the time constant \(\tau\) is a global parameter of the population or projection, ANNarchy can pre-compute the step size outside of the for loop over all neurons/synapses, which leads to huge increases in performance. The exponential method should therefore be reserved to first-order linear ODEs with the same time constant for all neurons/synapses:

neuron = Neuron(
    parameters = "tau = 10. : population",
    equations = "tau * dr/dt + r = sum(exc) : min=0.0, exponential"
)

2.13.1.4. Midpoint

The midpoint method is a Runge-Kutta method of order 2. It estimates the derivative in the middle of the interval \(t + \frac{h}{2}\).

\[ \begin{align}\begin{aligned}k_x = f(x(t), y(t))\\k_y = g(x(t), y(t))\\x(t+h) = x(t) + h \cdot f(x(t) + k_x \cdot \frac{h}{2}, y(t) + k_y \cdot \frac{h}{2})\\y(t+h) = y(t) + h \cdot g(x(t) + k_x \cdot \frac{h}{2}, y(t) + k_y \cdot \frac{h}{2})\end{aligned}\end{align} \]

2.13.1.5. Event-driven

Event-driven integration is only available for spiking synapses with variables following linear first-order dynamics. Let’s consider the following STDP synapse (see Spiking synapses for explanations):

STDP = Synapse(
    parameters = """
        tau_pre = 10.0 : postsynaptic
        tau_post = 10.0 : postsynaptic
    """,
    equations = """
        tau_pre * dApre/dt = - Apre : event-driven
        tau_post * dApost/dt = - Apost : event-driven
    """,
    pre_spike = """
        g_target += w
        Apre += cApre
        w = clip(w + Apost, 0.0 , 1.0)
    """,
    post_spike = """
        Apost += cApost
        w = clip(w + Apre, 0.0 , 1.0)
    """
)

The value of Apost and Apre is only needed when a pre- or post-synaptic spike occurs at the synapse, so there is no need to integrate the corresponding equations between two such events. First-order linear ODEs have the nice property that their analytical solution is easy to obtain. Let’s consider an equation of the form:

\[\tau \frac{dv}{dt} = E - v\]

If \(v\) has the value \(V_0\) at time \(t\), its value at time \(t + \Delta t\) is given by:

\[v(t + \Delta t) = V_0 \cdot \exp(-\frac{\Delta t}{\tau})\]

Note

If the synapse defines a psp argument (synaptic transmission is continuous), or if another continuous variable depends on the value of an event-driven one, it is not possible to use event-driven integration.

2.13.2. Order of evaluation

The values of variables are stored in a single array in order to save some memory. Special care therefore has to be taken on whether the update of a variable depends on the value of another variable at the previous time step or in the same step.

2.13.2.1. Systems of ODEs

Systems of ODEs are integrated concurrently, which means that the following system:

tau*dv/dt = I - v - u
tau*du/dt = v - u

would be numerized using the explicit Euler method as:

v[t+1] = v[t] + dt*(I - v[t] - u[t])/tau
u[t+1] = u[t] + dt*(v[t] - u[t])/tau

As we use a single array, the generated code is similar to:

new_v = v + dt*(I - v - u)/tau
new_u = u + dt*(v - u)/tau

v = new_v
u = new_u

This way, we ensure that the interdependent ODEs use the correct value for the other variables.

2.13.2.2. Assignments

When assignments (=, +=...) are used in an equations field, the order of valuation is different:

  • Assigments occurring before or after a system of ODEs are updated sequentially.
  • Systems of ODEs are updated concurrently.

Let’s consider the following dummy equations:

# Process the inputs
Exc = some_function(sum(exc))
Inh = another_function(sum(inh))
I = Exc - Inh
# ODE for the membrane potential, with a recovery variable
tau*dv/dt = I - v - u
tau*du/dt = v - u
# Firing rate is the positive part of v
r = pos(v)

Here, Exc and Inh represent the inputs to the neuron at the current time t. The new values should be immediately available for updating I, whose value should similarly be immediately used in the ODE of v. Similarly, the value of r should be the positive part of the value of v that was just calculated, not at the previous time step. Doing otherwise would introduce a lag in the neuron: changes in sum(exc) at t would be reflected in Exc at t+1, in I at t+2, in v at t+3 and finally in r at t+4. This is generally unwanted.

The generated code is therefore equivalent to:

# Process the inputs
Exc = some_function(sum(exc))
Inh = another_function(sum(inh))
I = Exc - Inh
# ODE for the membrane potential, with a recovery variable
new_v = v + dt*(I - v - u)/tau
new_u = u + dt*(v - u)/tau
v = new_v
u = new_u
# Firing rate is the positive part of v
r = pos(v)

One can even define multiple groups of assignments and systems of ODEs: systems of ODEs separated by at least one assignment will be evaluated sequentially (but concurrently inside each system). For example, in:

tau*du/dt = v - u
I = g_exc - g_inh
tau*dk/dt = v - k
tau*dv/dt = I - v - u + k

u and k are updated using the previous value of v, while v uses the new values of both I and u, but the previous one of k.