Group meeting brief: June 17, 2014

Justin's goal was to see if we can produce reverberatory bursts from the mean-field dynamics of the network model. There are two populations: neurons and synapses. The neurons follow integrate-and-fire model: $$\frac{d V}{dt} = - \frac{1}{\tau}(V - V_L) + A\sum_j Y_j (V - V_R)$$ while the synapses follow the TUM model with super inactive state $Q$ (TUMS): $$\frac{d X}{dt} = \frac{Q}{\tau_q} + \frac{Z}{\tau_r} - u X S$$ $$\frac{d Y}{dt} = - \frac{Y}{\tau_d} + u X S$$ $$\frac{d Z}{dt} = \frac{Y}{\tau_d} - \frac{Z}{\tau_r} - \frac{Z}{\tau_l}$$ $$\frac{d Q}{dt} = \frac{Z}{\tau_l} - \frac{Q}{\tau_s}$$ where $S$ is the spike chain of the presynaptic neuron. (That's why I changed the variable for super inactive state to $Q$.) The neuron distribution is one-dimensional on the membrane potential while the synapse distribution is 3 dimensional restricted by the constraint $X+Y+Z+Q=1$. The density functions $\rho$ obeys the continuity equation: $$\frac{\partial\rho}{\partial t} = - \mathbf{\nabla} \cdot \vec{J}$$ Basically, the current is given by the dynamic equations multiplying by the density. The mean-field quantities are given by the integrals like: $$\langle Y\rangle = \int Y \rho(Y, Z, Q) dv$$ with $dv = dY dZ dQ$. The time derivative becomes: $$\frac{d \langle Y\rangle}{dt} = - \int Y \mathbf{\nabla} \cdot \vec{J}$$ after substituting with the continuity equation and leads to $$\int dv J_Y$$ after partial integral. And, the mean-field equations look just like the original dynamical equations with variables replaced by their mean. Assuming the spike train is independent of the synaptic states, the spike term can be factorized: $$\langle u X S \rangle = u \langle X \rangle f$$ where $f = \langle S \rangle$ is the firing rate of the presynaptic neuron. The mean-field firing rate of a neuron can be calculated if we assume the active transmitter fraction $Y = \langle Y \rangle$ is constant or slow varying. The self-consistency of the mean-field analysis is complete when the neuronal and synaptic dynamics are combined.