Neurons, Nonlinear Models

Quadratic Integrate-and-Fire (QIF)

Formulation

\[ \tau_m \frac{dV_m(t)}{dt} = a \left(V_m(t) - V_\text{rest}\right)\left(V_m(t) - V_\text{crit}\right) + R_mI(t) \]

With approximation:

\[ V_m(t + \Delta t) \approx \frac{\Delta t}{\tau_m} \left[ a \left(V_m(t) - V_\text{rest}\right)\left(V_m(t) - V_\text{crit}\right) + R_mI(t) \right] + V_m(t) \]

After an action potential is generated:

\[V_m(t) \leftarrow V_\text{reset}\]

Where:

  • \(I\), total input current applied to the neuron \((\text{nA})\)

  • \(V_m\), electric potential difference across the cell membrane \((\text{mV})\)

  • \(V_\text{rest}\), equilibrium of the membrane potential \((\text{mV})\)

  • \(V_\text{crit}\), critical value of the membrane potential \((\text{mV})\)

  • \(a\), membrane potential tendency towards \(V_\text{rest}\) and away from \(V_\text{crit}\) (unitless)

  • \(V_\text{reset}\), membrane potential difference set after spiking \((\text{mV})\)

  • \(\tau_m\), membrane time constant \((\text{ms})\)

  • \(t\), current time of the simulation \((\text{ms})\)

  • \(\Delta t\), length of time over which each simulation step occurs \((\text{ms})\)

Under the conditions \(a > 0\) and \(V_\text{crit} > V_\text{rest}\).

Description

This model approximates exponential integrate-and-fire using a quadratic dynamics. It has two fixed points: the stable \(V_\text{rest}\) and unstable \(V_\text{crit}\). The rate at which the membrane voltage is attracted towards \(V_\text{rest}\) and repelled away from \(V_\text{crit}\) is controlled by \(a\).

Quadratic Integrate-and-Fire Slope Field of Membrane Voltage Quadratic Integrate-and-Fire Slope Field of Membrane Voltage

Slope field of the membrane voltage without any input current showing the relation between it and the critical voltage \((V_C = -50 \text{ mV})\) and rest voltage \((V_R = -60 \text{ mV})\) parameters. Plotted with values \(\tau_m=1 \text{ ms}\) and \(a=1\) over a time of \(1 \text{ ms}\).

References

  1. DOI:10.1017/CBO9781107447615 (§5.3)

Izhikevich (Adaptive Quadratic)

Formulation

\[\begin{split} \begin{align*} \tau_m \frac{dV_m(t)}{dt} &= a \left(V_m(t) - V_\text{rest}\right)\left(V_m(t) - V_\text{crit}\right) + R_mI(t) \\ I(t) &= I_x(t) - \sum_k w_k(t) \\ \tau_k \frac{dw_k(t)}{dt} &= b_k \left[ V_m(t) - V_\text{rest} \right] - w_k(t) \\ \end{align*} \end{split}\]

With approximations:

\[\begin{split} \begin{align*} V_m(t + \Delta t) &\approx \frac{\Delta t}{\tau_m} \left[ a \left(V_m(t) - V_\text{rest}\right)\left(V_m(t) - V_\text{crit}\right) + R_mI(t) \right] + V_m(t) \\ w_k(t + \Delta t) &\approx \frac{\Delta t}{\tau_k}\left[ b_k \left[ V_m(t) - V_\text{rest} \right] - w_k(t) \right] + w_k(t) \end{align*} \end{split}\]

After an action potential is generated:

\[\begin{split} \begin{align*} V_m(t) &\leftarrow V_\text{reset} \\ w_k(t) &\leftarrow w_k(t) + d_k \end{align*} \end{split}\]

Where:

  • \(I\), total input current applied to the neuron \((\text{nA})\)

  • \(I_x\), input current before adaptation \((\text{nA})\)

  • \(V_m\), electric potential difference across the cell membrane \((\text{mV})\)

  • \(V_\text{rest}\), equilibrium of the membrane potential \((\text{mV})\)

  • \(V_\text{crit}\), critical value of the membrane potential \((\text{mV})\)

  • \(a\), membrane potential tendency towards \(V_\text{rest}\) and away from \(V_\text{crit}\) (unitless)

  • \(V_\text{reset}\), membrane potential difference set after spiking \((\text{mV})\)

  • \(\tau_m\), membrane time constant \((\text{ms})\)

  • \(b_k\), subthreshold adaptation, voltage-current coupling \((\mu\text{S})\)

  • \(d_k\), spike-triggered current adaptation \((\text{nA})\)

  • \(\tau_k\), adaptation time constant \((\text{ms})\)

  • \(t\), current time of the simulation \((\text{ms})\)

  • \(\Delta t\), length of time over which each simulation step occurs \((\text{ms})\)

Under the conditions \(a > 0\) and \(V_\text{crit} > V_\text{rest}\).

Description

This model uses the same underlying dynamics of the quadratic integrate-and-fire neuron, but incorporates a linear adaptive current depeendent upon output spikes and the membrane voltage.

References

  1. DOI:10.1017/CBO9781107447615 (§6.1)

  2. DOI:10.3390/brainsci12070863

  3. DOI:10.7551/mitpress/2526.001.0001 (See §8.1.4)

Exponential Integrate-and-Fire (EIF)

Formulation

\[ \tau_m \frac{dV_m(t)}{dt} = - \left[V_m(t) - V_\text{rest}\right] + \Delta_T \exp \left(\frac{V_m(t) - V_T}{\Delta_T}\right) + R_mI(t) \]

With approximation:

\[ V_m(t + \Delta t) \approx \frac{\Delta t}{\tau_m} \left[ - \left[V_m(t) - V_\text{rest}\right] + \Delta_T \exp \left(\frac{V_m(t) - V_T}{\Delta_T}\right) + R_mI(t) \right]+ V_m(t) \]

After an action potential is generated:

\[V_m(t) \leftarrow V_\text{reset}\]

Where:

  • \(I\), total input current applied to the neuron \((\text{nA})\)

  • \(V_m\), electric potential difference across the cell membrane \((\text{mV})\)

  • \(V_\text{rest}\), equilibrium of the membrane potential \((\text{mV})\)

  • \(V_T\), membrane potential approaching the depolarization threshold \((\text{mV})\)

  • \(V_\text{reset}\), membrane potential difference set after spiking \((\text{mV})\)

  • \(\Delta_T\), steepness of depolarization and closeness to \(V_T\) \((\text{mV})\)

  • \(\tau_m\), membrane time constant \((\text{ms})\)

  • \(t\), current time of the simulation \((\text{ms})\)

  • \(\Delta t\), length of time over which each simulation step occurs \((\text{ms})\)

Under the conditions \(\Delta_T > 0\) and \(V_T > V_\text{rest}\).

Description

This model uses exponential dynamics to model the rapid increase in membrane voltage before (and at the start of) the generation of an action potential. This upswing occurs at a voltage above \(V_T\), specifically, as \(\Delta_T \rightarrow 0\), the voltage at which this upswing occurs approaches \(V_T\). \(\Delta_T\) also controls the sharpness of upswing.

Important

Two of the parameters used in EIF and its derivatives can be easy to confuse with unrelated parameters.

  • \(V_T\) is called the “threshold voltage” (although is occasionally called the rheobase threshold and denoted \(\vartheta_{rh}\)), but is different than the “voltage threshold”, i.e. the membrane potential at which an action potential is generated. The latter is often denoted as \(\Theta\), \(\Theta_\infty\), \(\theta\), or \(V_\text{thresh}\) and in Inferno is usually represented by the parameter thresh_v or thresh_eq_v.

  • \(\Delta_T\) is called the “slope factor” or “sharpness” and is unrelated to the “step time” used in discrete-time simuations of neuronal dynamics. The latter is typically denoted as \(\Delta t\) or \(\delta t\) and in Inferno is usually represented by the parameter step_time and attribute dt.

Below are two slope fields showing the relation between the threshold voltage \(V_T\) and rest voltage \(V_R\) in relation to the membrane voltage. Examples with two \(\Delta_T\) settings are used to illustrate its effect.

Exponential Integrate-and-Fire Slope Field of Membrane Voltage ($\Delta_T = 1$) Exponential Integrate-and-Fire Slope Field of Membrane Voltage ($\Delta_T = 1$)

Membrane voltage with no input current. Plotted with values \(V_R = -60\text{ mV}\), \(V_T = -50\), \(\Delta_T = 1\), and \(\tau_m=1 \text{ ms}\) over a time of \(1 \text{ ms}\).

Exponential Integrate-and-Fire Slope Field of Membrane Voltage ($\Delta_T = 2$) Exponential Integrate-and-Fire Slope Field of Membrane Voltage ($\Delta_T = 2$)

Membrane voltage with no input current. Plotted with values \(V_R = -60\text{ mV}\), \(V_T = -50\), \(\Delta_T = 2\), and \(\tau_m=1 \text{ ms}\) over a time of \(1 \text{ ms}\).

References

  1. DOI:10.1017/CBO9781107447615 (§6.1)

  2. ISBN:9780262548083

Adaptive Exponential Integrate-and-Fire (AdEx)

Formulation

\[\begin{split} \begin{align*} \tau_m \frac{dV_m(t)}{dt} &= - \left[V_m(t) - V_\text{rest}\right] + \Delta_T \exp \left(\frac{V_m(t) - V_T}{\Delta_T}\right) + R_mI(t) \\ I(t) &= I_x(t) - \sum_k w_k(t) \\ \tau_k \frac{dw_k(t)}{dt} &= a_k \left[ V_m(t) - V_\text{rest} \right] - w_k(t) \\ \end{align*} \end{split}\]

With approximations:

\[\begin{split} \begin{align*} V_m(t + \Delta t) &\approx \frac{\Delta t}{\tau_m} \left[- \left[V_m(t) - V_\text{rest}\right] + \Delta_T \exp \left(\frac{V_m(t) - V_T}{\Delta_T}\right) + R_mI(t) \right]+ V_m(t) \\ w_k(t + \Delta t) &\approx \frac{\Delta t}{\tau_k}\left[ a_k \left[ V_m(t) - V_\text{rest} \right] - w_k(t) \right] + w_k(t) \\ \end{align*} \end{split}\]

After an action potential is generated:

\[\begin{split} \begin{align*} V_m(t) &\leftarrow V_\text{reset} \\ w_k(t) &\leftarrow w_k(t) + b_k \end{align*} \end{split}\]

Where:

  • \(I\), total input current applied to the neuron \((\text{nA})\)

  • \(I_x\), input current before adaptation \((\text{nA})\)

  • \(V_m\), electric potential difference across the cell membrane \((\text{mV})\)

  • \(V_\text{rest}\), equilibrium of the membrane potential \((\text{mV})\)

  • \(V_T\), membrane potential approaching the depolarization threshold \((\text{mV})\)

  • \(V_\text{reset}\), membrane potential difference set after spiking \((\text{mV})\)

  • \(\Delta_T\), steepness of depolarization and closeness to \(V_T\) \((\text{mV})\)

  • \(\tau_m\), membrane time constant \((\text{ms})\)

  • \(a_k\), subthreshold adaptation, voltage-current coupling \((\mu\text{S})\)

  • \(b_k\), spike-triggered current adaptation \((\text{nA})\)

  • \(\tau_k\), adaptation time constant \((\text{ms})\)

  • \(t\), current time of the simulation \((\text{ms})\)

  • \(\Delta t\), length of time over which each simulation step occurs \((\text{ms})\)

Under the conditions \(\Delta_T > 0\) and \(V_T > V_\text{rest}\).

Description

This model uses the same underlying dynamics of the exponential integrate-and-fire neuron, but incorporates a linear adaptive current dependent upon output spikes and the membrane voltage.

References

  1. DOI:10.1017/CBO9781107447615 (§6.1)

  2. ISBN:9780262548083

  3. DOI:10.4249/scholarpedia.8427

  4. DOI:10.1523/JNEUROSCI.23-37-11628.2003