This is a summary of paper The Dynamic Brain: From Spiking Neurons to Neural Masses and Cortical Fields
This paper mainly talks about a variety of computational approaches that have been used to characterize the dynamics of the cortex.
- The central theme: the activity in populations of neurons can be understood by reducing the degrees of freedom from many to few.
- The way to achieve that is: to reduce the large population of spiking neurons to a distribution function describing their probabilistic evolution, which captures the likely distribution of neuronal states at a given time.
- This can be further reduced to a single variable describing the mean firing rate.
Neural Mass models: capture the dynamics of a neuronal population.
The theory based: Full probability distribution function can be represented by a set of scalars that parameterize it parsimoniously. These parameters are equivalent to the moments of the distribution.)
Neural Field Models: investigate how neuronal activity unfolds on the spatially continuous cortical sheet by involving differential operators with both temporal and spatial terms.
The theory based: Neuronal activity depends on its current state as well as spatial gradients, which allow its spread horizontally across the cortical surface.
Mean-Field Models
This part provides an overview of mean-field models of neuronal dynamics and their derivation from models of spiking neurons.
- mainly based on the mean-field approximation
- are formulated using concepts from statistical physics.
- suited to data which reflect the behavior of a population of neurons, such as EEG, MEG and fMRI.
Ensemble density models
- Ensemble models attempt to model the dynamics of large (theoretically infinite) populations of neurons
- use phase space to represent the neuron attribute space. Each attribute induces a dimension in the phase space of a neuron.
- Three attributes will be included: post-synaptic membrane depolarization, $V$, capacitive current, $I$, and the time since the last action potential, $T$. (therefore, it is a three dimensional phase space)
- The state of each neuron is a point $ν = {V,I,T} ∈ ℜ^3$
- The density of neurons populate the space can be represented as: $p(ν,t)$
- As the state of each neuron evolves, the points will flow through phase space, and the ensemble density $p(ν,t)$ will evolve until it reaches some steady state or equilibrium.
- $p(ν,t)$ is a scalar function returning the probability density at each point in phase space.
- the density dynamics conform to a simple equation: the Fokker-Planck equation:
- This equation comprises a flow and a dispersion term
- phase flow, $f(ν,t)$ describes the dynamics
- dispersion, $D(ν,t)$, describes the random fluctuations
$$\dot p = - \nabla \cdot (f-D\nabla)p \equiv \frac{\partial p}{\partial t} = tr(-\frac{\partial (fp)}{\partial v}+\frac{\partial}{\partial v}(D \frac {\partial p}{\partial v}))$$
- This level of description is usually framed as a stochastic differential equation (Langevin equation) that describes how the states evolve as functions of each other and some random fluctuations with $dv = f(v)dt + \sigma d \omega$.
- $D = \frac 12 \sigma^2$
- $\omega$ is a standard Wiener process,i.e. $w(t)−w(t+Δt)∼N(0, Δt)$
- The density dynamics can be written as a linear operator or Jacobian Q:
$$\dot p = Qp$$ $$Q = \nabla \cdot (D \nabla - f)$$ - For any model of neuronal dynamics, specified as a stochastic differential equation, there is a deterministic linear equation that can be integrated to generate ensemble dynamics.
From spiking neurons to mean-field models
- For a single neuron, we can assume the spiking dynamics as the $leaky integrate-and-fire (LIF)$ model.
- In the LIF model, each neuron i can be fully described in terms of a single internal variable, namely the depolarization $V_i(t)$ of the neural membrane.
- The basic circuit of a LIF model consists of a capacitor, $C$, in parallel with a resistor, $R$, driven by a synaptic current.
- When the voltage across the capacitor reaches a threshold $θ$, the circuit is shunted (reset) and a $δ$ pulse (spike) is generated and transmitted to other neurons.
- The subthreshold membrane potential of each neuron evolves according to a simple RC circuit, with a time constant $τ = RC$ given by the following equation:
- $I_i(t)$ is the total synaptic current flow into the cell i
- $V_L$ is the leak or resting potential of the cell in the absence of external afferent inputs
$$\tau \frac{dV_i(t)}{dt} = -[V_i(t)-V_L] + RI_i(t)$$
- The total synaptic current coming into the cell i is therefore given by the sum of the contributions of δ-spikes produced at presynaptic neurons.
- Assume that $N$ neurons synapse onto cell i and that $J_{ij}$ is the efficacy of synapse j, then the total synaptic afferent current is given by
- $t_j^{(k)}$ is the emission time of the kth spike from the jth presynaptic neuron.
$$RI_i(t) = \tau \sum^N_{j=1} J_{ij} \sum_k \delta (t-t_j^{(k)})$$
- $t_j^{(k)}$ is the emission time of the kth spike from the jth presynaptic neuron.
- Substitute the above equation gets:
- H(t) is the Heaviside function (H(t) = 1 if t>0, and H(t) = 0 if t<0) (acutally is a step function)
$$V_i(t) = V_L + \sum^N_{j=1} J_{ij} \int^t_0 e^{-s/\tau} \sum_k \delta (t-s-t_j^{(k)})ds
= V_L + \sum^N_{j=1} J_{ij} e^{-(t-t_j^{(k)})/\tau} \sum_k H(t-t_j^{(k)})$$
- H(t) is the Heaviside function (H(t) = 1 if t>0, and H(t) = 0 if t<0) (acutally is a step function)
The population density approach
A cortical column has $O(10^4)$−$O(10^8)$ neurons which are massively interconnected (on average, a neuron makes contact with $O(10^4)$ other neurons). The underlying dynamics of such networks can be described explicitly by the set of coupled differential equations. However, direct simulations of these equations will be very complex and computationally expensive. Therefore, we adopt the population density approach, using the Fokker-Planck formalism. The Fokker-Planck equation summarizes the flow and dispersion of states over phase space in a way that is a natural summary of population dynamics in genetics. The following will show how to derive the Fokker-Planck equation for neuronal dynamics.
- Individual IF neurons are grouped together into populations of statistically similar neurons.
- Then use probability density function to describe the distribution of neuronal states (i.e., membrane potential) over the population.
- Key assumption: the afferent input currents impinging on neurons in one population are uncorrelated
- In general, neurons with the same state $V(t)$ at a given time $t$ have a different history because of random fluctuations in the input current $I(t)$.
- The main source of randomness is from fluctuations in recurrent currents and fluctuations in the external currents
- Then the dynamics are described by the evolution of the probability density function:
- which is the fraction of neurons at time $t$ that have a membrane potential $V(t)$ in the interval $[ν,ν+dν]$
$$p(v, t)dv = Prob\{V(t)\in [v, v+dv]\}$$
- which is the fraction of neurons at time $t$ that have a membrane potential $V(t)$ in the interval $[ν,ν+dν]$
- The evolution of the population density is given by the Chapman-Kolmogorov equation:
- $ρ(ε|ν) = Prob\{V(t+dt) = ν+ε|V(t) = ν\}$ is the conditional probability that generates an infinitesimal change $ε = V(t+dt)−V(t)$ in the infinitesimal interval $dt$.
$$p(v, t+dt) = \int^{+\infty}_{-\infty} p(v-\varepsilon,t)\rho(\varepsilon|v-\varepsilon)d\varepsilon$$
- $ρ(ε|ν) = Prob\{V(t+dt) = ν+ε|V(t) = ν\}$ is the conditional probability that generates an infinitesimal change $ε = V(t+dt)−V(t)$ in the infinitesimal interval $dt$.
- The Chapman-Kolmogorov equation can be written in a differential form by performing a Taylor expansion in $p(ν′,t) ρ(ε|ν′)$ around $ν′ = ν$:
- assume that $p(ν′,t)$ and $ρ(ε| ν′)$ are infinitely many times differentiable in $ν$
$$p(v’,t)\rho (\varepsilon|v’) = \sum^{\infty}_{k=0} \frac{(-\varepsilon)^k}{k!} \frac{\partial^k}{\partial v’^k} [p(v’,t)\rho(\varepsilon|v’)] \mid {v’=v}$$
- assume that $p(ν′,t)$ and $ρ(ε| ν′)$ are infinitely many times differentiable in $ν$
- Combine the above equatin and can get:
- $〈…〉ν$ denotes the average with respect to $ρ(ε| ν)$ at a given $ν$
$$p(v,t+dt) = \sum^\infty_{k=0} \frac{(-1)^k}{k!} \frac{\partial^k}{\partial v^k} [p(v,t)\langle \varepsilon^k \rangle _v]$$
- $〈…〉ν$ denotes the average with respect to $ρ(ε| ν)$ at a given $ν$
- Take the limit for $dt \to 2$:
$$\frac{\partial p(v,t)}{\partial t} = \sum^\infty_{k=1} \frac{(-1)^k}{k!} \frac{\partial^k}{\partial v^k} [p(v,t)lim_{dt \to 0 }\frac 1{dt} \langle \varepsilon^k \rangle_v]$$
The diffusion approximation
- The above temporal evolution requires the moments $〈ε^k〉_υ$.
- These moments can be calculated by the mean-field approximation
- The mean-field approximation replaces the time-averaged discharge rate of individual cells with a common time-dependent population activity
- Therefore, infinitesimal change, $dV(t)$, in the membrane potential of all neurons is:
- $N$ is the number of neurons
- $〈J〉_J$ denotes the average of the synaptic weights in the population.
- $Q(t)$ is the mean population firing rate and determined by the proportion of active neurons by counting the number of spikes $n_{spikes}(t,t+dt)$ in a small time interval dt and dividing by N and by dt: $Q(t) = lim_{dt \to 0} \frac{n_{spikes}(t,t+dt)}{Ndt}$
$$dV(t) = \langle J\rangle_J NQ(t) dt - \frac{V(t)-V_L}{\tau}dt$$
- The first two moments in the Kramers-Moyal expansion are called drift and diffusion coefficients, respectively as follows:
$$M^{(1)} = lim_{dt \to 0} \frac1{dt} \langle \varepsilon\rangle_v =\langle J\rangle_J NQ(t) - \frac{v-V_L}{\tau} = \frac{\mu(t)}{\tau} - \frac{v-V_L}{\tau}$$ $$M^{(2)} = lim_{dt \to 0} \frac1{dt} \langle \varepsilon^2 \rangle_v =\langle J^2 \rangle_J NQ(t) = \frac{\sigma(t)^2}{\tau}$$ - The diffusion approximation allows to omit all higher orders k>2 in the Kramers-Moyal expansion. The resulting differential equation describing the temporal evolution of the population density is called the Fokker-Planck equation:
$$\frac{\partial p(v,t)}{\partial t} = \frac1{2\tau} \sigma^2 (t) \frac{\partial^2p(v,t)}{\partial v^2} + \frac{\partial}{\partial v}[(\frac{v-V_L-\mu(t)}{\tau})p(v,t)]$$ - If the drift is linear and the diffusion coefficient, $σ^2(t)$, is given by a constant, the Fokker-Planck equation describes a well-known stochastic process called the Ornstein-Uhlenbeck process. And the input afferent currents are given by
- $ω(t)$ is a white noise process
$$RI(t) = \mu(t) + \sigma \sqrt \tau \omega(t)$$
- $ω(t)$ is a white noise process