Subsections

• Equations
• Default values

   
PUM model equations

Equations

The temporal evolution of the potential and the threshold of NBC neurons in a network of N neurons takes place according to the following equations:

V_i(t)=U(t-T_pⁱ)+ $$\left[\vphantom{\sum_{j=1}^{N}(W_{ji}\int_{T_{p}^{i}}^{t}x_{j}(r - d_{ji})\,V_{psp}(t-r)\,dr) + \,B(t)}\right.$$$$\sum_{j=1}^{N}$$(W_ji$$\int_{T_{p}^{i}}^{t}$$x_j(r - d_ji) V_psp(t - r) dr) +  B(t)$$\left.\vphantom{\sum_{j=1}^{N}(W_{ji}\int_{T_{p}^{i}}^{t}x_{j}(r - d_{ji})\,V_{psp}(t-r)\,dr) + \,B(t)}\right]$$ m(t-T_pⁱ)

s_i(t) = F(t - T_pⁱ, s_i(T_pⁱ) +  S_A)

x_i(t) = $$\theta$$[V_i(t) - s_i(t)]

where

U(t)=U_rest+(U(0)-U_rest)e^{-t/$\tau_{V}$}

V_psp(t)=$$\left\{\vphantom{ \begin{array}{ll} V_{pspMax}\,(t/\tau_{attack})... ...-\tau_{attack}}{\tau_{decay}}} & when\;\tau_{attack}<t\\ \end{array} }\right.$$$$\begin{array}{ll} V_{pspMax}\,(t/\tau_{attack})\,e^{-\frac{t}{\ta... ...{-\frac{t-\tau_{attack}}{\tau_{decay}}} & when\;\tau_{attack}<t\\ \end{array}$$ $$\left.\vphantom{ \begin{array}{ll} V_{pspMax}\,(t/\tau_{attack})\... ...-\tau_{attack}}{\tau_{decay}}} & when\;\tau_{attack}<t\\ \end{array} }\right.$$

m(t)=1-e^{-t/$\tau_{shunt}$}

F(t, K) = s_rest + (K - s_rest)e^{-t/$\tau_{s}$}

$\theta$[y] is the Heaviside step function

$$\theta$$[y] = $$\left\{\vphantom{ \begin{array}{ll} 0 & if\;y < 0 \\ 1 & otherwise \end{array}}\right.$$$$\begin{array}{ll} 0 & if\;y < 0 \\ 1 & otherwise \end{array}$$ $$\left.\vphantom{ \begin{array}{ll} 0 & if\;y < 0 \\ 1 & otherwise \end{array}}\right.$$

U_rest,  U(0),  $\tau_{V}^{}$,  V_pspMax,  $\tau_{attack}^{}$,  $\tau_{decay}^{}$,  $\tau_{shunt}^{}$,  s_rest,  S_A,  $\tau_{s}^{}$ are the parameters which are set at the beginning of each simulation.

The threshold equation shows that the value of the threshold of a neuron at a given time tdepends on the firing history of the neuron. In fact by replacing F, the following expression for the threshold of a neuron which has fired pspikes, the kth spike happening at time T_k, is obtained:

s(t)=s_rest(1+e^{- ${\frac{t-T_{1}}{\tau_{s}}}$})+S_Ae^{- ${\frac{t}{\tau_{s}}}$} $$\sum_{k=1}^{p}$$e^{${\frac{T_{k}}{\tau_{s}}}$}

Default values

These are the default values of a Phenomenologic Model as stored in the .P_unit file:

ModelType: P
RestPotential: -70.0000
TimeCsteMbPot: 10.0000
Threshold: -65.0000
EpspSize: 0.5000
IpspSize: 0.0000
NoiseMean: 0.0000
NoiseSd: 0.0000
NoiseFilter: 0
Capacity: 1.0000000000
TimeStep: 1.000
PostSpikePotential: -85.0000
Adaptation: 20.0000
E_K: -90.0000
E_Na: 65.0000
PosReversalPot: 0.0000
NegReversalPot: -90.0000
TimeCsteThreshold: 20.0000
EpspSd: 0.0000
EpspAttack: 2.0000
EpspDecay: 5.0000
IpspSd: 0.0000
IpspAttack: 2.0000
IpspDecay: 5.0000
MembraneShunt: 20.0000
FisterResistance: 10.000
Fatigue: 0.000
LessFatigue: 0

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