Neurons, Nonlinear Models¶
Quadratic Integrate-and-Fire (QIF)¶
Formulation¶
With approximation:
After an action potential is generated:
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 \(a\).
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¶
Izhikevich (Adaptive Quadratic)¶
Formulation¶
With approximations:
After an action potential is generated:
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¶
Exponential Integrate-and-Fire (EIF)¶
Formulation¶
With approximation:
After an action potential is generated:
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
orthresh_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 attributedt
.
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.
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 \(1 \text{ ms}\).
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¶
Adaptive Exponential Integrate-and-Fire (AdEx)¶
Formulation¶
With approximations:
After an action potential is generated:
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.