Neuronal Dynamics (14)
Quasi-Renewal Theory and the Integral-equation Approach
The online version of this chapter:
Chapter 14 Quasi-Renewal Theory and the Integral-equation Approach https://neuronaldynamics.epfl.ch/online/Ch14.html
For neuron models that include biophysical phenomena such as refractoriness and adaptation on multiple time-scales, the resulting PDEs are situated in more than 2D and therefore difficult to solve analytically. We indicate an alternative to describing the population activity in networks of model neurons.
Population activity equations
Assumptions of time-dependent renewal theory
The state of a neuron \(i\) at time \(t\) is described by - its last firing time \(\hat{t}_i\); - the input \(I(t')\) it received for times \(t'<t\); - the characteristics of potential noise sources, be it noise in the input, noise in the neuronal parameters, or noise in the output.
Integral equations for non-adaptive neurons
The integral equation for activity dynamics with time-dependent renewal theory states that \[ A(t)=\int_{-\infty}^{t} P_{I}(t|\hat{t})A(\hat{t}) \mathrm{d}\hat{t}. \tag{14.5} \]
\(A(t)\) on the LHS is the expected activity at time \(t\) while \(A(\hat{t})\) on the RHS is the observed activity in the past. (14.5) becomes exact in the limit of \(N \to \infty\). Finite-size effects are discussed later.
- (14.5) is linear in the variable \(A\). Instead of defining the activity by a spike count divided by \(N\), we could have chosen to work directly with the spike count per unit of time or any other normalization.
- (14.5) is a highly nonlinear equation in the drive because the kernel \(P_{I}(t|\hat{t})\) depends nonlinearly on the input \(I(t)\).
Normalization and derivation of the integral equation
Recap \[ S_{I}(t|\hat{t})=1-\int_{\hat{t}}^{t} P_{I}(s|\hat{t}) \mathrm{d}s, \tag{14.6} \]
We now return to the homogeneous population of neurons in the limit of \(N \to \infty\) and assume that the firing of different neurons at time \(t\) is independent, given that we know the history of each neuron. ('conditional independence')
Define the proportion of neurons at time \(t\) which have fired their last spike between \(t_0\) and \(t_1<t\) (and have not firied since) as \[ \left\langle \frac{\text{number of neurons at $t$ with last spike in}\ [t_0,t_1]}{\text{total number of neurons}} \right\rangle =\int_{t_0}^{t_1} S_{I}(t|\hat{t})A(\hat{t}) \mathrm{d}\hat{t}. \tag{14.7} \]
We use the fact that the total number of neurons remains constant. \[ 1=\int_{-\infty}^{t} S_{I}(t|\hat{t})A(\hat{t}) \mathrm{d}\hat{t}, \tag{14.8} \] The normalization of (14.8) must hold at arbitrary times \(t\).
Take the derivative of (14.8) w.r.t \(t\), \[ 0=S_{I}(t|t)A(t)+\int_{-\infty}^{t} \frac{\partial S_{I}(t|\hat{t})}{\partial t}A(\hat{t}) \mathrm{d}\hat{t}. \tag{14.9} \]
Use \(P_{I}(t|\hat{t})=-\frac{\partial }{\partial t}S_{I}(t|\hat{t})\) and \(S_{I}(t|t)=1\) and we yield (14.5).
Example: Absolute refractoriness and the Wilson-Cowan integral equation
Consider a population of Poisson neurons with an absolute refractory period \(\Delta^{abs}\).
The population activity of a homogeneous group of Poisson neurons with absolute refractoriness is \[ A(t)=f[h(t)] \left\{ 1-\int_{t-\Delta^{abs}}^{t} A(t') \mathrm{d}t' \right\}. \tag{14.10} \] where \(h(t)=\int_{0}^{\infty} \kappa(s)I(t-s) \mathrm{d}s\). \(f\) is the stochastic intensity of an inhomogeneous Poisson process describing neurons in a homogeneous population.
For constant input current, \(h(t)=h_0=I_0 \int_{0}^{\infty} \kappa(s) \mathrm{d}s\). The population activity has a stationary solution \[ A_0=\frac{f(h_0)}{1+\Delta^{abs}f(h_0)}=g(h_0). \tag{14.11} \]
Integral equation for adaptive neurons
For an isolated adaptive neuron in the presence of noise, the probability density of firing around time \(t\) will depend on its past firing times \(\hat{t}_{n}<\cdots <\hat{t}_2<\hat{t}_1=\hat{t}<t\) where \(\hat{t}=\hat{t}_1\) denotes the most recent spike time.
In a population of neurons, we can approximate the past firing times by the population activity \(A(t)\). Let \(P_{I,A}(t|\hat{t})\) be the probability of observing a spike at time \(t\) given the last spike at time \(\hat{t}\), the input-current and the activity history \(A(t)\) until time \(t\), then \[ A(t)=\int_{-\infty}^{t} P_{I,A}(t|\hat{t})A(\hat{t}) \mathrm{d}\hat{t}. \tag{14.12} \]
Numerical methods for integral equations
We take as an example the quasi-renewal equivalent of (14.8) \[ 1=\int_{-\infty}^{t} S_{I,A}(t|\hat{t})A(\hat{t}) \mathrm{d}\hat{t}. \tag{14.13} \]
Let \(\tau_c\) be a period time such that the survivor \(S_{I,A}(t|\hat{t-\tau_c})\) is very small. Then \[ 1=\int_{t-\tau_c}^{t} S_{I,A}(t|\hat{t})A(\hat{t}) \mathrm{d}\hat{t}. \tag{14.14} \]
Discretize the integral on small bins of size \(\Delta t\). Let \(\mathbf{m}^{(t)}\) be the vector made of the fraction of neurons at \(t\) with last spike within \(\hat{t}\) and \(\hat{t}+\Delta t\), which means that the \(k\)th element is \(m_k^{(t)}=S(t|t-k\Delta t)A(t-k\Delta t)\Delta t\), \(m_0^{(t)}=A_t\Delta t\) since \(S(t|t)=1\). Therefore \[ A_t \Delta t=1-\sum_{k=1}^{K} m_k^{(t)}. \tag{14.15} \]
Because of \(S(t|\hat{t})=\exp [-\int_{\hat{t}}^{t} \rho(t'|\hat{t}) \mathrm{d}t']=\exp [-\int_{t-\Delta t}^{t} \rho(t'|\hat{t}) \mathrm{d}t']S(t-\Delta t|\hat{t})\), we find for sufficiently small \(\Delta t\), \[ m_k^{(t)}=m_{k-1}^{(t-\Delta t)}\exp [-\rho(t|t-k\Delta t)\Delta t] \quad\text{for}\ k\geqslant 1 \tag{14.16} \]
Note that \(m_k^{(t)}\) and \(m_{k-1}^{(t-\Delta t)}\) refer to the same group of neurons, i.e., those that have fired their last spike around time \(\hat{t}=t-k\Delta t\). Together, (14.15) and (14.16) can be used to solve \(A_t\) iteratively.