Tutorial to build simple networks of synaptically coupled neurons. The membrane potential dynamics are obtained using the thermodynamic transmembrane transport by Herrera-Valdez (PeerJ Preprints, 2017)
Neuronal dynamics in small networks: Simple synaptic coupling
Marco Arieli Herrera-Valdez, Fernando Pérez Díaz, Roberto García-Medina, José Alberto Perez Benitez
Facultad de Ciencias, Universidad Nacional Autónoma de México
ESIME-SEPI, Instituto Politécnico Nacional
Consider a general minimal model for transmembrane potential for a single cell where Auxiliary functions
The amplitude of the synaptic current is described by a sum of an Ornstein-Uhlenbeck process representing non-specific inputs and a sum of synaptic currents that depend on the activity of the other cells. The parameters , , , represent, respectively, the half-activation potential, gating charge, de-activation bias, and basal rate for activation of the gate represented by the variable .
Simple network with plasticity
with auxiliary functions ParseError: KaTeX parse error: Undefined control sequence: \label at position 222: …,u,E \right\}, \̲l̲a̲b̲e̲l̲{uSS} \\ R(v,v_…
\paragraph{Synapses between PINs} Let the connection matrix be where ParseError: KaTeX parse error: Undefined control sequence: \lrSet at position 12: w_{ij} \in \̲l̲r̲S̲e̲t̲{0,1} represents the connection weight from neuron to neuron . For simplification purposes, the activation of the presynaptic terminals in the th neuron is assumed to depend on the transmembrane potential of the membrane potential of the th neuron.
The excitatory postsynaptic current for the th neuron is given by where
Evolution of the system assuming that synapses activate as functions of time
Funciones auxiliares
Solución numérica de ecuaciones
Change of variables to simplify the numerics
Let and . Then As a consequence, the amplitudes in the original system can also be normalized:
Numerical method: Fourth order Runge-Kutta
Parameters
Calculation of the solution
Initial conditions and number of steps
The solution to the system is composed of two vectors that correspond to the two variables of the system, and . Then y can be transformed back into by setting .
Network dynamics with simple synaptic coupling
The activation of synapses depends only on the voltage of the presynaptic neuron. Assume for simplicity that there are neurons in the network. Construct two matrices for the connections such that the first neuron excites the second, and the second neuron excites the first:
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NameError Traceback (most recent call last)
<ipython-input-2-731127457a27> in <module>()
----> 1 pars["nCells"]=2
2 v0=-sc.ones(pars["nCells"])*30.0/pars["v_T"]
3 y0= sc.ones(pars["nCells"])
4 pars["ic"]= sc.array([v0,y0])
5 pars["timeMax"] = 40.0
NameError: name 'pars' is not defined