Model library#

Note

Dendrify relies on Brian’s Equations-based approach to define models as systems of first order ordinary differential equations. For convenience, Dendrify includes a library of default models (see below) however users can also provide custom model equations.

Somatic models [1] [2]#

Leaky Integrate-and-Fire#

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

where \(C\) is the membrane capacitance, \(V\) the membrane potential, \(g_L\) the leak conductance, \(E_L\) the leak reversal potential and \(I\) is the input current. When the firing threshold \(V_\theta\) is crossed, \(V\) resets to a fixed value \(V_r\).

Adaptive Integrate-and-Fire#

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

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

where \(w\) is the adaptation variable, \(a\) the adaptation coupling parameter and \(τ_w\) is the adaptation time constant. When the firing threshold \(V_\theta\) is crossed, \(V\) resets to a fixed value \(V_r\) and \(w \rightarrow w+b\), where \(b\) is the spike-triggered adaptation current.

Adaptive Exponential Integrate-and-Fire#

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

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

where \(\Delta_T\) is the slope factor and \(V_T\) the voltage threshold. When the firing threshold \(V_\theta\) is crossed, \(V\) resets to a fixed value \(V_r\) and \(w \rightarrow w+b\), where \(b\) is the spike-triggered adaptation current.

Dendritic models#

Synaptic models [3] [4]#

AMPA#

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

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

where \(\bar{g}_{\text{AMPA}}\) is the AMPA synaptic conductance, \(s\) the channel state variable, \(E_{\text{AMPA}}\) the AMPA reversal potential, \(V\) the membrane potential and \(\tau_{\text{AMPA}}^{\text{decay}}\) the AMPA decay time constant. When a pre-synaptic spike arrives \(s \rightarrow s+1\).

AMPA (rise & decay)#

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

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

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

where \(s\) and \(x\) describe the rise and decay kinetics of the channel respectively, \(\tau_{\text{AMPA}}^{\text{rise}}\) is the AMPA rise time constant and \(\tau_{\text{AMPA}}^{\text{decay}}\) is the AMPA decay time constant. When a pre-synaptic spike arrives \(s \rightarrow s+1\).

NMDA#

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

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

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

where \(\bar{g}_{\text{NMDA}}\) is the NMDA synaptic conductance, \(s\) the channel state variable, \(E_{\text{NMDA}}\) the NMDA reversal potential, \(\tau_{\text{NMDA}}^{\text{decay}}\) the NMDA decay time constant, \(\beta\) (mM), \(\alpha\) (mV^-1) and \(\gamma\) (mV) control the magnesium and voltage dependencies and \([\rm{Mg}^{2+}]_{o}\) denotes the external magnesium concentration (mM). When a pre-synaptic spike arrives \(s \rightarrow s+1\).