Quadratic integrate and fire

The quadratic integrate and fire (QIF) model is a <a href="/facts/Biological_neuron_model/eCPeWaXc">biological neuron model</a> that describes <a href="/facts/Action_potentials/Sz8yCRqx">action potentials</a> in neurons. In contrast to physiologically accurate but computationally expensive neuron models like the <a href="/facts/Hodgkin%25E2%2580%2593Huxley_model/HMuK0gP9">Hodgkin–Huxley model</a>, the QIF model seeks only to produce action potential-like patterns by ignoring the dynamics of transmembrane currents and <a href="/facts/Ion_channels/A46LARFH">ion channels</a>. Thus, the QIF model is computationally efficient and has found ubiquitous use in computational neuroscience.
An idealized model of neural spiking is given by the <a href="/facts/Autonomous_differential_equation/ahlr3A6o">autonomous differential equation</a>,

d
              x
            
            
              d
              t
            
          
        
        =
        
          x
          
            2
          
        
        +
        I
      
    
    {\displaystyle {\frac {dx}{dt}}=x^{2}+I}

where 
 
 
 
 x
 
 
 {\displaystyle x}
 
 represents the membrane voltage and 
 
 
 
 I
 ≥
 0
 
 
 {\displaystyle I\geq 0}
 
 represents an input current. A solution to this differential equation is the function,

x
        (
        t
        )
        =
        
          
            I
          
        
        tan
        ⁡
        (
        
          
            I
          
        
        t
        +
        
          c
          
            0
          
        
        )
        ,
      
    
    {\displaystyle x(t)={\sqrt {I}}\tan({\sqrt {I}}t+c_{0}),}

where 
 
 
 
 
 c
 
 0
 
 
 
 
 {\displaystyle c_{0}}
 
 is an arbitrary shift dependent on the initial condition 
 
 
 
 x
 (
 0
 )
 
 
 {\displaystyle x(0)}
 
 (specifically by the formula 
 
 
 
 
 c
 
 0
 
 
 =
 arctan
 ⁡
 (
 x
 (
 0
 )
 
 /
 
 
 
 I
 
 
 )
 
 
 {\displaystyle c_{0}=\arctan(x(0)/{\sqrt {I}})}
 
). This solution "blows up" in finite time, namely at 
 
 
 
 t
 =
 n
 π
 
 /
 
 (
 2
 
 
 I
 
 
 )
 −
 
 c
 
 0
 
 
 
 
 {\displaystyle t=n\pi /(2{\sqrt {I}})-c_{0}}
 
 for all 
 
 
 
 n
 ∈
 
 N
 
 
 
 {\displaystyle n\in \mathbb {N} }
 
, which resembles the rhythmic action potentials generated by neurons stimulated by some input current. Thus a "spike" is said to have occurred when the solution reaches positive infinity. Just after this point in time, the solution resets to negative infinity by definition.
When implementing this model in a numerical simulation, a threshold crossing value (
 
 
 
 
 V
 
 t
 
 
 
 
 {\displaystyle V_{t}}
 
) and a reset value (
 
 
 
 
 V
 
 r
 
 
 
 
 {\displaystyle V_{r}}
 
) is assigned, so that when the solution rises above the threshold, 
 
 
 
 x
 (
 t
 )
 ≥
 
 V
 
 t
 
 
 
 
 {\displaystyle x(t)\geq V_{t}}
 
, the solution is immediately reset to 
 
 
 
 
 V
 
 r
 
 
 
 
 {\displaystyle V_{r}}
 
.
The above equation is directly related to an alternative form of the QIF model,

d
 v
 
 
 d
 t
 
 
 
 =
 −
 
 
 
 v
 (
 1
 −
 v
 )
 
 
 τ
 
 m
 
 
 
 
 +
 I
 
 
 {\displaystyle {\frac {dv}{dt}}=-{\frac {v(1-v)}{\tau _{m}}}+I}
 
,
where 
 
 
 
 
 τ
 
 m
 
 
 
 
 {\displaystyle \tau _{m}}
 
 is the membrane time constant.

Quadratic integrate and fire open-in-new

Quadratic integrate and fire