Basic Models in Theoretical Neuroscience Oren Shriki 2010 Synaptic Dynamics 1

t t_spike A spike at time t spike of the presynaptic cell contributes to the postsynaptic cell a time-dependent conductance, g s (t): Synaptic Conductances Peak conductance 2

Synaptic Dynamics: An Example Synaptic dynamics are usually characterized by fast rise and slow decay. The simplest model assumes instantaneous rise and exponential decay: (Presynaptic rate) 3

Synaptic Dynamics: An Example For a single presynaptic spike the solution is: 4

Synaptic Dynamics: An Example Implementation in numerical simulations: Given the time step dt define the attenuation factor: A dimensionless parameter, f, is increased by 1 after each presynaptic spike and multiplied by the attenuation factor in each time step. The conductance is the product of f and the peak conductance, G. 5

Synaptic Dynamics For simplicity, we shall write in general: K(t) is the time course (dimensionless) function. We define: For example: 6

Network Architecture External Inputs Recurrent connectivity 1234N 7

Voltage Dynamics for A Network of Conductance-Based Point Neurons We assume that the neurons are point neurons obeying Hodgkin-Huxley type dynamics: I active – Ionic current involved in the action potential I ext – External synaptic inputs I net – Synaptic inputs from within the network I app – External current applied by the experimentalist 8

External Synaptic Current The explicit expression for the external synaptic current is: The peak synaptic conductance is: The time constant is: 9

Internal Synaptic Current The explicit expression for the internal synaptic current is: The peak synaptic conductance is: The time constant is: 10