# 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 $a$. ```{image} ../images/plots/qif-slope-field-light.png :alt: Quadratic Integrate-and-Fire Slope Field of Membrane Voltage :class: only-light :width: 30em :align: center ``` ```{image} ../images/plots/qif-slope-field-dark.png :alt: Quadratic Integrate-and-Fire Slope Field of Membrane Voltage :class: only-dark :width: 30em :align: center ``` 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)](https://neuronaldynamics.epfl.ch/online/Ch5.S3.html) ## Izhikevich (Adaptive Quadratic) ### Formulation $$ \begin{align*} \tau_m \frac{dV_m(t)}{dt} &= a \left(V_m(t) - V_\text{rest}\right)\left(V_m(t) - V_\text{crit}\right) + R_mI(t) \\ I(t) &= I_x(t) - \sum_k w_k(t) \\ \tau_k \frac{dw_k(t)}{dt} &= b_k \left[ V_m(t) - V_\text{rest} \right] - w_k(t) \\ \end{align*} $$ *With approximations:* $$ \begin{align*} V_m(t + \Delta t) &\approx \frac{\Delta t}{\tau_m} \left[ a \left(V_m(t) - V_\text{rest}\right)\left(V_m(t) - V_\text{crit}\right) + R_mI(t) \right] + V_m(t) \\ w_k(t + \Delta t) &\approx \frac{\Delta t}{\tau_k}\left[ b_k \left[ V_m(t) - V_\text{rest} \right] - w_k(t) \right] + w_k(t) \end{align*} $$ *After an action potential is generated:* $$ \begin{align*} V_m(t) &\leftarrow V_\text{reset} \\ w_k(t) &\leftarrow w_k(t) + d_k \end{align*} $$ *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)](https://neuronaldynamics.epfl.ch/online/Ch6.S1.html) 1. [DOI:10.3390/brainsci12070863](https://www.mdpi.com/2076-3425/12/7/863/pdf) 1. [DOI:10.7551/mitpress/2526.001.0001 (See §8.1.4)](https://www.izhikevich.org/publications/dsn.pdf) ## 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. ```{image} ../images/plots/eif-slope-field-d1-light.png :alt: Exponential Integrate-and-Fire Slope Field of Membrane Voltage ($\Delta_T = 1$) :class: only-light :width: 30em :align: center ``` ```{image} ../images/plots/eif-slope-field-d1-dark.png :alt: Exponential Integrate-and-Fire Slope Field of Membrane Voltage ($\Delta_T = 1$) :class: only-dark :width: 30em :align: center ``` 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 $1 \text{ ms}$. ```{image} ../images/plots/eif-slope-field-d2-light.png :alt: Exponential Integrate-and-Fire Slope Field of Membrane Voltage ($\Delta_T = 2$) :class: only-light :width: 30em :align: center ``` ```{image} ../images/plots/eif-slope-field-d2-dark.png :alt: Exponential Integrate-and-Fire Slope Field of Membrane Voltage ($\Delta_T = 2$) :class: only-dark :width: 30em :align: center ``` 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)](https://neuronaldynamics.epfl.ch/online/Ch6.S1.html) 1. [ISBN:9780262548083](https://mitpress.ublish.com/ebook/modeling-neural-circuits-made-simple-with-python-preview/12788/Cover) ## Adaptive Exponential Integrate-and-Fire (AdEx) ### Formulation $$ \begin{align*} \tau_m \frac{dV_m(t)}{dt} &= - \left[V_m(t) - V_\text{rest}\right] + \Delta_T \exp \left(\frac{V_m(t) - V_T}{\Delta_T}\right) + R_mI(t) \\ I(t) &= I_x(t) - \sum_k w_k(t) \\ \tau_k \frac{dw_k(t)}{dt} &= a_k \left[ V_m(t) - V_\text{rest} \right] - w_k(t) \\ \end{align*} $$ *With approximations:* $$ \begin{align*} V_m(t + \Delta t) &\approx \frac{\Delta t}{\tau_m} \left[- \left[V_m(t) - V_\text{rest}\right] + \Delta_T \exp \left(\frac{V_m(t) - V_T}{\Delta_T}\right) + R_mI(t) \right]+ V_m(t) \\ w_k(t + \Delta t) &\approx \frac{\Delta t}{\tau_k}\left[ a_k \left[ V_m(t) - V_\text{rest} \right] - w_k(t) \right] + w_k(t) \\ \end{align*} $$ *After an action potential is generated:* $$ \begin{align*} V_m(t) &\leftarrow V_\text{reset} \\ w_k(t) &\leftarrow w_k(t) + b_k \end{align*} $$ *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)](https://neuronaldynamics.epfl.ch/online/Ch6.S1.html) 1. [ISBN:9780262548083](https://mitpress.ublish.com/ebook/modeling-neural-circuits-made-simple-with-python-preview/12788/Cover) 1. [DOI:10.4249/scholarpedia.8427](http://www.scholarpedia.org/article/Adaptive_exponential_integrate-and-fire_model) 1. [DOI:10.1523/JNEUROSCI.23-37-11628.2003](https://www.jneurosci.org/content/23/37/11628.long)