Model library#

Leaky membrane#

\[C\dfrac{dV}{dt}=-g_L(V-E_L)+I\]

Symbol	Description
\(C\)	membrane capacitance
\(V\)	membrane potential
\(g_L\)	leakage conductance
\(E_L\)	leakage reversal potential
\(I\)	input current

Somatic spiking models#

Leaky Integrate-and-Fire#

\[C\dfrac{dV}{dt}=-g_L(V-E_L)+I\]

Spike mechanism:

\[\text{if } V \geq V_\theta \text{ then } V \rightarrow V_r\]

Symbol	Description
\(V_\theta\)	spike threshold
\(V_r\)	reset potential

Examples:

Adaptive Integrate-and-Fire#

\[C\dfrac{dV}{dt}=-g_L(V-E_L)-w+I\]

\[\tau_w\dfrac{dw}{dt}=a(V-E_L)-w\]

Spike mechanism:

\[\begin{split}\text{if } V \geq V_\theta \text{ then } \begin{cases} V \rightarrow V_r \\ w \rightarrow w + b \end{cases}\end{split}\]

Symbol	Description
\(w\)	adaptation current
\(a\)	maximal adaptation conductance
\(b\)	spike-triggered adaptation current
\(τ_w\)	adaptation time constant
\(V_\theta\)	spike threshold
\(V_r\)	reset potential

Examples:

Other adaptive models

Conductance-based Adaptive Integrate-and-Fire#

\[C\dfrac{dV}{dt}=-g_L(V-E_L)-w+I\]

\[w=g_A(V-E_A)\]

\[\tau_A\dfrac{dg_A}{dt}=\bar{g_A}|V-E_A|\gamma - g_A\]

Spike mechanism:

\[\begin{split}\text{if } V \geq V_\theta \text{ then } \begin{cases} V \rightarrow V_r \\ g_A \rightarrow g_A + \delta g_A \end{cases}\end{split}\]

Symbol	Description
\(w\)	adaptation current
\(\bar{g_A}\)	maximal adaptation conductance
\(E_A\)	reversal potential of the adaptation
\(τ_A\)	adaptation time constant
\(\delta g_A\)	spike-triggered adaptation conductance
\(\gamma\)	steepness of the adaptation
\(V_\theta\)	spike threshold
\(V_r\)	reset potential

Examples:

Other adaptive models

Adaptive Exponential Integrate-and-Fire#

\[C\dfrac{dV}{dt}=-g_L(V-E_L)+g_L\Delta_T\exp\left(\dfrac{V-V_T}{\Delta_T}\right)-w+I\]

\[\tau_w\dfrac{dw}{dt}=a(V-E_L)-w\]

Spike mechanism:

\[\begin{split}\text{if } V \geq V_\theta \text{ then } \begin{cases} V \rightarrow V_r \\ w \rightarrow w + b \end{cases}\end{split}\]

Symbol	Description
\(w\)	adaptation current
\(a\)	maximal adaptation conductance
\(b\)	spike-triggered adaptation current
\(V_T\)	voltage threshold
\(\Delta_T\)	slope factor
\(τ_w\)	adaptation time constant
\(V_\theta\)	effective spike threshold
\(V_r\)	reset potential

Examples:

Synapses#

AMPA#

\[I_{\text{AMPA}}=\bar{g}_{\text{AMPA}}(E_{\text{AMPA}}-V)s(t)\]

\[\dfrac{ds}{dt}=\dfrac{-s}{\tau_{\text{AMPA}}^{\text{decay}}}\]

At presynaptic firing time:

\[s \rightarrow s+1\]

Symbol	Description
\(\bar{g}_{\text{AMPA}}\)	maximal AMPA conductance
\(E_{\text{AMPA}}\)	AMPA reversal potential
\(\tau_{\text{AMPA}}^{\text{decay}}\)	AMPA decay time constant
\(s\)	channel state variable
\(V\)	membrane potential

Examples:

AMPA (rise & decay)#

\[I_{\text{AMPA}}=\bar{g}_{\text{AMPA}}(E_{\text{AMPA}}-V)x(t)\]

\[\dfrac{dx}{dt}=\dfrac{-x}{\tau_{\text{AMPA}}^{\text{decay}}}+s(t)\]

\[\dfrac{ds}{dt}=\dfrac{-s}{\tau_{\text{AMPA}}^{\text{rise}}}\]

At presynaptic firing time:

\[s \rightarrow s+1\]

Symbol	Description
\(\bar{g}_{\text{AMPA}}\)	maximal AMPA conductance
\(E_{\text{AMPA}}\)	AMPA reversal potential
\(\tau_{\text{AMPA}}^{\text{decay}}\)	AMPA decay time constant
\(s\)	rise state variable
\(x\)	decay state variable
\(V\)	membrane potential

Examples:

AMPA synapses

NMDA#

\[I_{\text{NMDA}}=\bar{g}_{\text{NMDA}}(E_{\text{NMDA}}-V)s(t)\sigma(V)\]

\[\dfrac{ds}{dt}=\dfrac{-s}{\tau_{\text{NMDA}}^{\text{decay}}}\]

\[\sigma(V)=\dfrac{1}{1+\dfrac{{\left[{\rm{Mg}}^{2+}\right]}_{o}}{\beta }\cdot {{\exp }}\left(-\alpha \left(V-\gamma \right)\right)}\]

At presynaptic firing time:

\[s \rightarrow s+1\]

Symbol	Description
\(\bar{g}_{\text{NMDA}}\)	maximal NMDA conductance
\(E_{\text{NMDA}}\)	NMDA reversal potential
\(\tau_{\text{NMDA}}^{\text{decay}}\)	NMDA decay time constant
\(s\)	channel state variable
\(\alpha\)	the steepness of Magnesium unblock
\(\beta\)	the sensitivity of Magnesium unblock
\(\gamma\)	offset of the Magnesium unblock
\([\rm{Mg}^{2+}]_{o}\)	external Magnesium concentration

Examples:

AdEx network + synapses

NMDA (rise & decay)#

\[I_{\text{NMDA}}=\bar{g}_{\text{NMDA}}(E_{\text{NMDA}}-V)x(t)\sigma(V)\]

\[\dfrac{dx}{dt}=\dfrac{-x}{\tau_{\text{NMDA}}^{\text{decay}}}+s(t)\]

\[\dfrac{ds}{dt}=\dfrac{-s}{\tau_{\text{NMDA}}^{\text{rise}}}\]

\[\sigma(V)=\dfrac{1}{1+\dfrac{{\left[{\rm{Mg}}^{2+}\right]}_{o}}{\beta }\cdot {{\exp }}\left(-\alpha \left(V-\gamma \right)\right)}\]

At presynaptic firing time:

\[s \rightarrow s+1\]

Symbol	Description
\(\bar{g}_{\text{NMDA}}\)	maximal NMDA conductance
\(E_{\text{NMDA}}\)	NMDA reversal potential
\(\tau_{\text{NMDA}}^{\text{decay}}\)	NMDA decay time constant
\(s\)	rise state variable
\(x\)	decay state variable
\(\alpha\)	the steepness of Magnesium unblock
\(\beta\)	the sensitivity of Magnesium unblock
\(\gamma\)	offset of the Magnesium unblock
\([\rm{Mg}^{2+}]_{o}\)	external Magnesium concentration

Examples:

NMDA synapses

GABA#

\[I_{\text{GABA}}=\bar{g}_{\text{GABA}}(E_{\text{GABA}}-V)s(t)\]

\[\dfrac{ds}{dt}=\dfrac{-s}{\tau_{\text{GABA}}^{\text{decay}}}\]

At presynaptic firing time:

\[s \rightarrow s+1\]

Symbol	Description
\(\bar{g}_{\text{GABA}}\)	maximal GABA conductance
\(E_{\text{GABA}}\)	GABA reversal potential
\(\tau_{\text{GABA}}^{\text{decay}}\)	GABA decay time constant
\(s\)	channel state variable
\(V\)	membrane potential

Examples:

GABA (rise & decay)#

\[I_{\text{GABA}}=\bar{g}_{\text{GABA}}(E_{\text{GABA}}-V)x(t)\]

\[\dfrac{dx}{dt}=\dfrac{-x}{\tau_{\text{GABA}}^{\text{decay}}}+s(t)\]

\[\dfrac{ds}{dt}=\dfrac{-s}{\tau_{\text{GABA}}^{\text{rise}}}\]

At presynaptic firing time:

\[s \rightarrow s+1\]

Symbol	Description
\(\bar{g}_{\text{GABA}}\)	maximal GABA conductance
\(E_{\text{GABA}}\)	GABA reversal potential
\(\tau_{\text{GABA}}^{\text{decay}}\)	GABA decay time constant
\(s\)	rise state variable
\(x\)	decay state variable
\(V\)	membrane potential

Examples:

NMDA synapses

Study material#

https://neuronaldynamics.epfl.ch/online/Ch1.html

https://neuronaldynamics.epfl.ch/online/Ch3.html

https://link.springer.com/chapter/10.1007/978-0-387-87708-2_2

https://link.springer.com/chapter/10.1007/978-0-387-87708-2_7#Sec1