Spike response model

<h2 id="model-equations-for-srm-in-continuous-time">Model equations for SRM in continuous time</h2>
In the SRM, at each moment in time t, a spike can be generated stochastically with instantaneous stochastic intensity or 'escape function'<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a><a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

ρ
        (
        t
        )
        =
        f
        (
        V
        (
        t
        )
        −
        ϑ
        (
        t
        )
        )
      
    
    {\displaystyle \rho (t)=f(V(t)-\vartheta (t))}

that depends on the momentary difference between the membrane voltage V(t) and the dynamic threshold 
 
 
 
 ϑ
 (
 t
 )
 
 
 {\displaystyle \vartheta (t)}
 
.
The membrane voltage V(t) at time t is given by<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a><a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>

V
        (
        t
        )
        =
        
          ∑
          
            f
          
        
        η
        (
        t
        −
        
          t
          
            f
          
        
        )
        +
        
          ∫
          
            0
          
          
            ∞
          
        
        κ
        (
        s
        )
        I
        (
        t
        −
        s
        )
        
        d
        s
        +
        
          V
          
            
              r
              e
              s
              t
            
          
        
      
    
    {\displaystyle V(t)=\sum _{f}\eta (t-t^{f})+\int _{0}^{\infty }\kappa (s)I(t-s)\,ds+V_{\mathrm {rest} }}

where tf is the firing time of spike number f of the neuron, Vrest is the resting voltage in the absence of input, I(t-s) is the input current at time t − s and 
 
 
 
 κ
 (
 s
 )
 
 
 {\displaystyle \kappa (s)}
 
 is a linear filter (also called kernel) that describes the contribution of an input current pulse at time t − s to the voltage at time t. The contributions to the voltage caused by a spike at time 
 
 
 
 
 t
 
 f
 
 
 
 
 {\displaystyle t^{f}}
 
 are described by the refractory kernel 
 
 
 
 η
 (
 t
 −
 
 t
 
 f
 
 
 )
 
 
 {\displaystyle \eta (t-t^{f})}
 
. In particular,

 
 
 
 η
 (
 t
 −
 
 t
 
 f
 
 
 )
 
 
 {\displaystyle \eta (t-t^{f})}
 
 describes the time course of the action potential starting at time 
 
 
 
 
 t
 
 f
 
 
 
 
 {\displaystyle t^{f}}
 
 as well as the spike-afterpotential.
The dynamic threshold 
 
 
 
 ϑ
 (
 t
 )
 
 
 {\displaystyle \vartheta (t)}
 
 is given by<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a><a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>

ϑ
        (
        t
        )
        =
        
          ϑ
          
            0
          
        
        +
        
          ∑
          
            f
          
        
        
          θ
          
            1
          
        
        (
        t
        −
        
          t
          
            f
          
        
        )
      
    
    {\displaystyle \vartheta (t)=\vartheta _{0}+\sum _{f}\theta _{1}(t-t^{f})}

where 
 
 
 
 
 ϑ
 
 0
 
 
 
 
 {\displaystyle \vartheta _{0}}
 
 is the firing threshold of an inactive neuron and 
 
 
 
 
 θ
 
 1
 
 
 (
 t
 −
 
 t
 
 f
 
 
 )
 
 
 {\displaystyle \theta _{1}(t-t^{f})}
 
 describes the increase of the threshold after a spike at time 
 
 
 
 
 t
 
 f
 
 
 
 
 {\displaystyle t^{f}}
 
. In case of a fixed threshold [i.e., 
 
 
 
 
 θ
 
 1
 
 
 (
 t
 −
 
 t
 
 f
 
 
 )
 
 
 {\displaystyle \theta _{1}(t-t^{f})}
 
=0], the refractory kernel 
 
 
 
 η
 (
 t
 −
 
 t
 
 f
 
 
 )
 
 
 {\displaystyle \eta (t-t^{f})}
 
 should include only the spike-afterpotential, but not the shape of the spike itself.
A common choice<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a><a class="footnote-ref" id="fnref:21" href="#fn:21">21</a><a class="footnote-ref" id="fnref:22" href="#fn:22">22</a> for the 'escape rate' 
 
 
 
 f
 
 
 {\displaystyle f}
 
 (that is consistent with biological data<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a>) is

f
        (
        V
        −
        ϑ
        )
        =
        
          
            1
            
              τ
              
                0
              
            
          
        
        exp
        ⁡
        [
        β
        (
        V
        −
        ϑ
        )
        ]
      
    
    {\displaystyle f(V-\vartheta )={\frac {1}{\tau _{0}}}\exp[\beta (V-\vartheta )]}

where 
 
 
 
 
 τ
 
 0
 
 
 
 
 {\displaystyle \tau _{0}}
 
 is a time constant that describes how quickly a spike is fired once the membrane potential reaches the threshold and 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
 is a sharpness parameter. For 
 
 
 
 β
 →
 ∞
 
 
 {\displaystyle \beta \to \infty }
 
 the threshold becomes sharp and spike firing occurs deterministically at the moment when the membrane potential hits the threshold from below. The sharpness value found in experiments is 
 
 
 
 1
 
 /
 
 β
 ≈
 4
 m
 V
 
 
 {\displaystyle 1/\beta \approx 4mV}
 
 which that neuronal firing becomes non-neglibable as soon the membrane potential is a few mV below the formal firing threshold. The escape rate process via a soft threshold is reviewed in Chapter 9 of the textbook Neuronal Dynamics.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a>
In a network of N SRM neurons 
 
 
 
 1
 ≤
 i
 ≤
 N
 
 
 {\displaystyle 1\leq i\leq N}
 
, the membrane voltage of neuron 
 
 
 
 i
 
 
 {\displaystyle i}
 
 is given by<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a>

V
          
            i
          
        
        (
        t
        )
        =
        
          ∑
          
            f
          
        
        
          η
          
            i
          
        
        (
        t
        −
        
          t
          
            i
          
          
            f
          
        
        )
        +
        
          ∑
          
            j
            =
            1
          
          
            N
          
        
        
          w
          
            i
            j
          
        
        
          ∑
          
            
              f
              ′
            
          
        
        
          ε
          
            i
            j
          
        
        (
        t
        −
        
          t
          
            j
          
          
            
              f
              ′
            
          
        
        )
        +
        
          V
          
            
              r
              e
              s
              t
            
          
        
      
    
    {\displaystyle V_{i}(t)=\sum _{f}\eta _{i}(t-t_{i}^{f})+\sum _{j=1}^{N}w_{ij}\sum _{f'}\varepsilon _{ij}(t-t_{j}^{f'})+V_{\mathrm {rest} }}

where 
 
 
 
 
 t
 
 j
 
 
 
 f
 ′
 
 
 
 
 
 {\displaystyle t_{j}^{f'}}
 
 are the firing times of neuron j (i.e., its spike train), and 
 
 
 
 
 η
 
 i
 
 
 (
 t
 −
 
 t
 
 i
 
 
 f
 
 
 )
 
 
 {\displaystyle \eta _{i}(t-t_{i}^{f})}
 
 describes the time course of the spike and the spike after-potential for neuron i, 
 
 
 
 
 w
 
 i
 j
 
 
 
 
 {\displaystyle w_{ij}}
 
 and 
 
 
 
 
 ε
 
 i
 j
 
 
 (
 t
 −
 
 t
 
 j
 
 
 
 f
 ′
 
 
 
 )
 
 
 {\displaystyle \varepsilon _{ij}(t-t_{j}^{f'})}
 
 describe the amplitude and time course of an excitatory or inhibitory postsynaptic potential (PSP) caused by the spike 
 
 
 
 
 t
 
 j
 
 
 
 f
 ′
 
 
 
 
 
 {\displaystyle t_{j}^{f'}}
 
 of the presynaptic neuron j. The time course 
 
 
 
 
 ε
 
 i
 j
 
 
 (
 s
 )
 
 
 {\displaystyle \varepsilon _{ij}(s)}
 
 of the PSP results from the convolution of the postsynaptic current 
 
 
 
 I
 (
 t
 )
 
 
 {\displaystyle I(t)}
 
 caused by the arrival of a presynaptic spike from neuron j.

<h2 id="model-equations-for-srm-in-discrete-time">Model equations for SRM in discrete time</h2>
For simulations, the SRM is usually implemented in discrete time.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a><a class="footnote-ref" id="fnref:27" href="#fn:27">27</a> In time step 
 
 
 
 
 t
 
 n
 
 
 
 
 {\displaystyle t_{n}}
 
 of duration 
 
 
 
 Δ
 t
 
 
 {\displaystyle \Delta t}
 
, a spike is generated with probability

P
          
            F
          
        
        (
        
          t
          
            n
          
        
        )
        =
        F
        (
        V
        (
        
          t
          
            n
          
        
        )
        −
        ϑ
        (
        
          t
          
            n
          
        
        )
        )
      
    
    {\displaystyle P_{F}(t_{n})=F(V(t_{n})-\vartheta (t_{n}))}

that depends on the momentary difference between the membrane voltage V and the dynamic threshold 
 
 
 
 ϑ
 
 
 {\displaystyle \vartheta }
 
. The function F is often taken as a standard sigmoidal 
 
 
 
 F
 (
 x
 )
 =
 0.5
 [
 1
 +
 tanh
 ⁡
 (
 γ
 x
 )
 ]
 
 
 {\displaystyle F(x)=0.5[1+\tanh(\gamma x)]}
 
<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a> with steepness parameter 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
. But the functional form of F can also be calculated from the stochastic intensity 
 
 
 
 f
 
 
 {\displaystyle f}
 
 in continuous time as 
 
 
 
 F
 (
 
 y
 
 n
 
 
 )
 ≈
 1
 −
 exp
 ⁡
 [
 
 y
 
 n
 
 
 
 Δ
 t
 ]
 
 
 {\displaystyle F(y_{n})\approx 1-\exp[y_{n}\,\Delta t]}
 
 where 
 
 
 
 
 y
 
 n
 
 
 =
 V
 (
 
 t
 
 n
 
 
 )
 −
 ϑ
 (
 
 t
 
 n
 
 
 )
 
 
 {\displaystyle y_{n}=V(t_{n})-\vartheta (t_{n})}
 
 is the distance to threshold.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a><a class="footnote-ref" id="fnref:30" href="#fn:30">30</a>
The membrane voltage 
 
 
 
 V
 (
 
 t
 
 n
 
 
 )
 
 
 {\displaystyle V(t_{n})}
 
 in discrete time is given by

V
        (
        
          t
          
            n
          
        
        )
        =
        
          ∑
          
            f
          
        
        η
        (
        
          t
          
            n
          
        
        −
        
          t
          
            f
          
        
        )
        +
        
          ∑
          
            m
            =
            1
          
          
            ∞
          
        
        κ
        (
        m
        
        Δ
        t
        )
        I
        (
        
          t
          
            n
          
        
        −
        m
        
        Δ
        t
        )
        +
        
          V
          
            
              r
              e
              s
              t
            
          
        
      
    
    {\displaystyle V(t_{n})=\sum _{f}\eta (t_{n}-t^{f})+\sum _{m=1}^{\infty }\kappa (m\,\Delta t)I(t_{n}-m\,\Delta t)+V_{\mathrm {rest} }}

where tf is the discretized firing time of the neuron, Vrest is the resting voltage in the absence of input, and 
 
 
 
 I
 (
 
 t
 
 k
 
 
 )
 
 
 {\displaystyle I(t_{k})}
 
 is the input current at time 
 
 
 
 
 t
 
 k
 
 
 
 
 {\displaystyle t_{k}}
 
 (integrated over one time step). The input filter 
 
 
 
 κ
 (
 s
 )
 
 
 {\displaystyle \kappa (s)}
 
 and the spike-afterpotential 
 
 
 
 η
 (
 s
 )
 
 
 {\displaystyle \eta (s)}
 
 are defined as in the case of the SRM in continuous time.
For networks of SRM neurons in discrete time we define the spike train of neuron j as a sequence of zeros and ones, 
 
 
 
 {
 
 X
 
 j
 
 
 (
 
 t
 
 m
 
 
 )
 ∈
 {
 0
 ,
 1
 }
 ;
 m
 =
 1
 ,
 2
 ,
 3
 ,
 …
 }
 
 
 {\displaystyle \{X_{j}(t_{m})\in \{0,1\};m=1,2,3,\dots \}}
 
 and rewrite the membrane potential as<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a>

V
          
            i
          
        
        (
        
          t
          
            n
          
        
        )
        =
        
          ∑
          
            m
          
        
        
          η
          
            i
          
        
        (
        
          t
          
            n
          
        
        −
        
          t
          
            m
          
        
        )
        
          X
          
            i
          
        
        (
        
          t
          
            m
          
        
        )
        +
        
          ∑
          
            j
          
        
        
          w
          
            i
            j
          
        
        
          ∑
          
            m
          
        
        
          ε
          
            i
            j
          
        
        (
        
          t
          
            n
          
        
        −
        
          t
          
            m
          
        
        )
        
          X
          
            j
          
        
        (
        
          t
          
            m
          
        
        )
        +
        
          V
          
            
              r
              e
              s
              t
            
          
        
      
    
    {\displaystyle V_{i}(t_{n})=\sum _{m}\eta _{i}(t_{n}-t_{m})X_{i}(t_{m})+\sum _{j}w_{ij}\sum _{m}\varepsilon _{ij}(t_{n}-t_{m})X_{j}(t_{m})+V_{\mathrm {rest} }}

In this notation, the refractory kernel 
 
 
 
 κ
 (
 s
 )
 
 
 {\displaystyle \kappa (s)}
 
 and the PSP shape 
 
 
 
 
 ε
 
 i
 j
 
 
 (
 s
 )
 
 
 {\displaystyle \varepsilon _{ij}(s)}
 
 can be interpreted as linear response filters applied to the binary spike trains 
 
 
 
 
 X
 
 j
 
 
 
 
 {\displaystyle X_{j}}
 
.

<h2 id="main-applications-of-the-srm">Main applications of the SRM</h2>
<h3>Theory of computation with pulsed neural networks</h3>
Since the formulation as SRM provides an explicit expression for the membrane voltage (without the detour via a differential equations), SRMs have been the dominant mathematical model in a formal theory of computation with spiking neurons.<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a><a class="footnote-ref" id="fnref:33" href="#fn:33">33</a><a class="footnote-ref" id="fnref:34" href="#fn:34">34</a>

<h3>Prediction of voltage and spike times of cortical neurons</h3>
The SRM with dynamic threshold has been used to predict the firing time of cortical neurons with a precision of a few milliseconds.<a class="footnote-ref" id="fnref:35" href="#fn:35">35</a> Neurons were stimulated, via current injection, with time-dependent currents of different means and variance while the membrane voltage was recorded. The reliability of predicted spikes was close to the intrinsic reliability when the same time-dependent current was repeated several times. Moreover, extracting the shape of the filters 
 
 
 
 κ
 (
 s
 )
 
 
 {\displaystyle \kappa (s)}
 
 and 
 
 
 
 η
 (
 s
 )
 
 
 {\displaystyle \eta (s)}
 
 directly from the experimental data revealed that adaptation extends over time scales from tens of milliseconds to tens of seconds.<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a><a class="footnote-ref" id="fnref:37" href="#fn:37">37</a><a class="footnote-ref" id="fnref:38" href="#fn:38">38</a> Thanks to the convexity properties of the likelihood in Generalized Linear Models,<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a><a class="footnote-ref" id="fnref:40" href="#fn:40">40</a> parameter extraction is efficient.<a class="footnote-ref" id="fnref:41" href="#fn:41">41</a>

<h3>Associative memory in networks of spiking neurons</h3>
SRM0 neurons have been used to construct an associative memory in a network of spiking neurons.<a class="footnote-ref" id="fnref:42" href="#fn:42">42</a><a class="footnote-ref" id="fnref:43" href="#fn:43">43</a> The SRM network<a class="footnote-ref" id="fnref:44" href="#fn:44">44</a> which stored a finite number of stationary patterns as attractors using a Hopfield-type connectivity matrix<a class="footnote-ref" id="fnref:45" href="#fn:45">45</a> was one of the first examples of attractor networks with spiking neurons.<a class="footnote-ref" id="fnref:46" href="#fn:46">46</a><a class="footnote-ref" id="fnref:47" href="#fn:47">47</a><a class="footnote-ref" id="fnref:48" href="#fn:48">48</a>

<h3>Population activity equations in large networks of spiking neurons</h3>
For SRM neurons, an important variable characterizing the internal state of the neuron is the time since the last spike (or 'age' of the neuron) which enters into the refractory kernel 
 
 
 
 η
 (
 s
 )
 
 
 {\displaystyle \eta (s)}
 
. The population activity equations for SRM neurons can be formulated alternatively either as integral equations,<a class="footnote-ref" id="fnref:49" href="#fn:49">49</a><a class="footnote-ref" id="fnref:50" href="#fn:50">50</a><a class="footnote-ref" id="fnref:51" href="#fn:51">51</a> or as partial differential equations for the 'refractory density'.<a class="footnote-ref" id="fnref:52" href="#fn:52">52</a><a class="footnote-ref" id="fnref:53" href="#fn:53">53</a> Because the refractory kernel may include a time scale slower than that of the membrane potential, the population equations for SRM neurons provide powerful alternatives<a class="footnote-ref" id="fnref:54" href="#fn:54">54</a><a class="footnote-ref" id="fnref:55" href="#fn:55">55</a><a class="footnote-ref" id="fnref:56" href="#fn:56">56</a> to the more broadly used partial differential equations for the 'membrane potential density'.<a class="footnote-ref" id="fnref:57" href="#fn:57">57</a><a class="footnote-ref" id="fnref:58" href="#fn:58">58</a><a class="footnote-ref" id="fnref:59" href="#fn:59">59</a> Reviews of the population activity equation based on refractory densities can be found in<a class="footnote-ref" id="fnref:60" href="#fn:60">60</a> as well in Chapter 14 of the textbook Neuronal Dynamics.<a class="footnote-ref" id="fnref:61" href="#fn:61">61</a>

<h3>Spike patterns and temporal code</h3>
SRMs are useful to understand theories of <a href="/facts/Neural_coding/GSRJ4WJF">neural coding</a>. A network SRM neurons has stored attractors that form reliable spatio-temporal spike patterns<a class="footnote-ref" id="fnref:62" href="#fn:62">62</a> (also known as synfire chains<a class="footnote-ref" id="fnref:63" href="#fn:63">63</a>) example of temporal coding for stationary inputs. Moreover, the population activity equations for SRM exhibit temporally precise transients after a stimulus switch, indicating reliable spike firing.<a class="footnote-ref" id="fnref:64" href="#fn:64">64</a>

<h2 id="history-and-relation-to-other-models">History and relation to other models</h2>
The Spike Response Model has been introduced in a series of papers between 1991<a class="footnote-ref" id="fnref:65" href="#fn:65">65</a> and 2000.<a class="footnote-ref" id="fnref:66" href="#fn:66">66</a><a class="footnote-ref" id="fnref:67" href="#fn:67">67</a><a class="footnote-ref" id="fnref:68" href="#fn:68">68</a><a class="footnote-ref" id="fnref:69" href="#fn:69">69</a> The name Spike Response Model probably appeared for the first time in 1993.<a class="footnote-ref" id="fnref:70" href="#fn:70">70</a> Some papers used exclusively the deterministic limit with a hard threshold<a class="footnote-ref" id="fnref:71" href="#fn:71">71</a> others the soft threshold with escape noise.<a class="footnote-ref" id="fnref:72" href="#fn:72">72</a> Precursors of the Spike Response Model are the <a href="/facts/Integrate-and-fire_model/eCPeWaXc">integrate-and-fire model</a> introduced by Lapicque in 1907 as well as models used in auditory neuroscience.<a class="footnote-ref" id="fnref:73" href="#fn:73">73</a><a class="footnote-ref" id="fnref:74" href="#fn:74">74</a><a class="footnote-ref" id="fnref:75" href="#fn:75">75</a><a class="footnote-ref" id="fnref:76" href="#fn:76">76</a>

<h3>SRM0</h3>
An important variant of the model is SRM0<a class="footnote-ref" id="fnref:77" href="#fn:77">77</a> which is related to time-dependent nonlinear <a href="/facts/Renewal_theory/1rDc2czB">renewal theory</a>. The main difference to the voltage equation of the SRM introduced above is that in the term containing the refractory kernel 
 
 
 
 η
 (
 s
 )
 
 
 {\displaystyle \eta (s)}
 
 there is no summation sign over past spikes: only the most recent spike matters. The model SRM0 is closely related to the inhomogeneous Markov interval process<a class="footnote-ref" id="fnref:78" href="#fn:78">78</a> and to age-dependent models of refractoriness.<a class="footnote-ref" id="fnref:79" href="#fn:79">79</a><a class="footnote-ref" id="fnref:80" href="#fn:80">80</a><a class="footnote-ref" id="fnref:81" href="#fn:81">81</a>

<h3>GLM</h3>
The equations of the SRM as introduced above are equivalent to Generalized Linear Models in neuroscience (GLM).<a class="footnote-ref" id="fnref:82" href="#fn:82">82</a><a class="footnote-ref" id="fnref:83" href="#fn:83">83</a> In the neuroscience, GLMs have been introduced as an extension of the <a href="/facts/Linear-nonlinear-Poisson_cascade_model/HSyfsN30">Linear-Nonlinear-Poisson</a> model (LNP) by adding self-interaction of an output spike with the internal state of the neuron<a class="footnote-ref" id="fnref:84" href="#fn:84">84</a><a class="footnote-ref" id="fnref:85" href="#fn:85">85</a> (therefore also called 'Recursive LNP'). The self-interaction is equivalent to the kernel 
 
 
 
 η
 (
 s
 )
 
 
 {\displaystyle \eta (s)}
 
 of the SRM. The GLM framework enables to formulate a maximum likelihood approach<a class="footnote-ref" id="fnref:86" href="#fn:86">86</a> applied to the <a href="https://neuronaldynamics.epfl.ch/online/Ch9.S2.html">likelihood of an observed spike train</a> under the assumption that an SRM could have generated the spike train.<a class="footnote-ref" id="fnref:87" href="#fn:87">87</a> Despite the mathematical equivalence there is a conceptual difference in interpretation: in the SRM the variable V is interpreted as membrane voltage whereas in the recursive LNP it is a 'hidden' variable to which no meaning is assigned. The SRM interpretation is useful if measurements of subthreshold voltage are available<a class="footnote-ref" id="fnref:88" href="#fn:88">88</a><a class="footnote-ref" id="fnref:89" href="#fn:89">89</a><a class="footnote-ref" id="fnref:90" href="#fn:90">90</a> whereas the recursive LNP is useful in systems neuroscience where spikes (in response to sensory stimulation) are recorded extracellulary without access to the subthreshold voltage.<a class="footnote-ref" id="fnref:91" href="#fn:91">91</a><a class="footnote-ref" id="fnref:92" href="#fn:92">92</a>

<h3>Adaptive leaky integrate-and-fire models</h3>
A leaky <a href="/facts/Biological_neuron_model/eCPeWaXc">integrate-and-fire neuron</a> with spike-triggered adaptation has a subthreshold membrane potential generated by the following differential equations

τ
          
            
              m
            
          
        
        
          
            
              d
              V
              (
              t
              )
            
            
              d
              t
            
          
        
        =
        R
        I
        (
        t
        )
        −
        [
        V
        (
        t
        )
        −
        
          E
          
            
              r
              e
              s
              t
            
          
        
        ]
        −
        R
        
          ∑
          
            k
          
        
        
          w
          
            k
          
        
      
    
    {\displaystyle \tau _{\mathrm {m} }{\frac {dV(t)}{dt}}=RI(t)-[V(t)-E_{\mathrm {rest} }]-R\sum _{k}w_{k}}

τ
          
            k
          
        
        
          
            
              d
              
                w
                
                  k
                
              
              (
              t
              )
            
            
              d
              t
            
          
        
        =
        −
        
          w
          
            k
          
        
        +
        
          b
          
            k
          
        
        
          τ
          
            k
          
        
        
          ∑
          
            f
          
        
        δ
        (
        t
        −
        
          t
          
            f
          
        
        )
      
    
    {\displaystyle \tau _{k}{\frac {dw_{k}(t)}{dt}}=-w_{k}+b_{k}\tau _{k}\sum _{f}\delta (t-t^{f})}

where 
 
 
 
 
 τ
 
 m
 
 
 
 
 {\displaystyle \tau _{m}}
 
 is the membrane time constant and wk is an adaptation current number, with index k, Erest is the resting potential and tf is the firing time of the neuron and the Greek delta denotes the Dirac delta function. Whenever the voltage reaches the firing threshold the voltage is reset to a value Vr below the firing threshold. Integration of the linear differential equations gives a formula identical to the voltage equation of the SRM.<a class="footnote-ref" id="fnref:93" href="#fn:93">93</a> However, in this case, the refractory kernel 
 
 
 
 η
 (
 s
 )
 
 
 {\displaystyle \eta (s)}
 
 does not include the spike shape but only the spike-afterpotential. In the absence of adaptation currents, we retrieve the standard LIF model which is equivalent to a refractory kernel 
 
 
 
 η
 (
 s
 )
 
 
 {\displaystyle \eta (s)}
 
 that decays exponentially with the membrane time constant 
 
 
 
 
 τ
 
 m
 
 
 
 
 {\displaystyle \tau _{m}}
 
.

<h2 id="external-links">External links</h2>
<ul><li><a href="https://neuronaldynamics.epfl.ch/online/Ch6.S4.html">Spike Response Model</a>, Chapter 6.4 of the textbook Neuronal Dynamics<a class="footnote-ref" id="fnref:94" href="#fn:94">94</a></li>
<li><a href="https://neuronaldynamics.epfl.ch/online/Ch9.html">'soft threshold' and escape noise</a>, Chapter 9 of the textbook Neuronal Dynamics<a class="footnote-ref" id="fnref:95" href="#fn:95">95</a></li>
<li><a href="https://neuronaldynamics.epfl.ch/online/Ch14.html">Quasi-Renewal Theory</a> Chapter 14 of the textbook Neuronal Dynamics.<a class="footnote-ref" id="fnref:96" href="#fn:96">96</a></li>
<li><a href="http://scholarpedia.org/article/Spike-response_model">Spike Response Model</a>, from Scholarpedia<a class="footnote-ref" id="fnref:97" href="#fn:97">97</a></li></ul>
<h2 id="reference-section">Reference section</h2>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Gerstner, Wulfram; Ritz, Raphael; van Hemmen, J. Leo (1993-10-01). "Why spikes? Hebbian learning and retrieval of time-resolved excitation patterns". Biological Cybernetics. 69 (5): 503–515. doi:10.1007/BF00199450. ISSN 1432-0770. PMID 7903867. S2CID 6195748. <a href="https://doi.org/10.1007/BF00199450" target="_blank">https://doi.org/10.1007/BF00199450</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Gerstner, Wulfram; van Hemmen, J. Leo; Cowan, Jack D. (1996-11-15). "What Matters in Neuronal Locking?". Neural Computation. 8 (8): 1653–1676. doi:10.1162/neco.1996.8.8.1653. ISSN 0899-7667. PMID 8888612. S2CID 1301248. <a href="https://doi.org/10.1162/neco.1996.8.8.1653" target="_blank">https://doi.org/10.1162/neco.1996.8.8.1653</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Gerstner, Wulfram; Ritz, Raphael; van Hemmen, J. Leo (1993-10-01). "Why spikes? Hebbian learning and retrieval of time-resolved excitation patterns". Biological Cybernetics. 69 (5): 503–515. doi:10.1007/BF00199450. ISSN 1432-0770. PMID 7903867. S2CID 6195748. <a href="https://doi.org/10.1007/BF00199450" target="_blank">https://doi.org/10.1007/BF00199450</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Gerstner, Wulfram; van Hemmen, J. Leo; Cowan, Jack D. (1996-11-15). "What Matters in Neuronal Locking?". Neural Computation. 8 (8): 1653–1676. doi:10.1162/neco.1996.8.8.1653. ISSN 0899-7667. PMID 8888612. S2CID 1301248. <a href="https://doi.org/10.1162/neco.1996.8.8.1653" target="_blank">https://doi.org/10.1162/neco.1996.8.8.1653</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Maass, Wolfgang (1997-02-01). "Fast Sigmoidal Networks via Spiking Neurons". Neural Computation. 9 (2): 279–304. doi:10.1162/neco.1997.9.2.279. ISSN 0899-7667. PMID 9117904. S2CID 34984298. <a href="https://doi.org/10.1162/neco.1997.9.2.279" target="_blank">https://doi.org/10.1162/neco.1997.9.2.279</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Jolivet, Renaud; Rauch, Alexander; Lüscher, Hans-Rudolf; Gerstner, Wulfram (August 2006). "Predicting spike timing of neocortical pyramidal neurons by simple threshold models". Journal of Computational Neuroscience. 21 (1): 35–49. doi:10.1007/s10827-006-7074-5. ISSN 0929-5313. PMID 16633938. S2CID 8911457. <a href="http://link.springer.com/10.1007/s10827-006-7074-5" target="_blank">http://link.springer.com/10.1007/s10827-006-7074-5</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Gerstner, Wulfram; Hemmen, J. Leo van (1992-01-01). "Associative memory in a network of 'spiking' neurons". Network: Computation in Neural Systems. 3 (2): 139–164. doi:10.1088/0954-898X_3_2_004. ISSN 0954-898X. <a href="https://doi.org/10.1088/0954-898X_3_2_004" target="_blank">https://doi.org/10.1088/0954-898X_3_2_004</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Truccolo, Wilson; Eden, Uri T.; Fellows, Matthew R.; Donoghue, John P.; Brown, Emery N. (2005-02-01). "A Point Process Framework for Relating Neural Spiking Activity to Spiking History, Neural Ensemble, and Extrinsic Covariate Effects". Journal of Neurophysiology. 93 (2): 1074–1089. doi:10.1152/jn.00697.2004. ISSN 0022-3077. PMID 15356183. <a href="https://journals.physiology.org/doi/full/10.1152/jn.00697.2004" target="_blank">https://journals.physiology.org/doi/full/10.1152/jn.00697.2004</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Pillow, Jonathan W.; Shlens, Jonathon; Paninski, Liam; Sher, Alexander; Litke, Alan M.; Chichilnisky, E. J.; Simoncelli, Eero P. (August 2008). "Spatio-temporal correlations and visual signalling in a complete neuronal population". Nature. 454 (7207): 995–999. Bibcode:2008Natur.454..995P. doi:10.1038/nature07140. ISSN 1476-4687. PMC 2684455. PMID 18650810. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2684455" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2684455</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Gerstner, Wulfram; van Hemmen, J. Leo; Cowan, Jack D. (1996-11-15). "What Matters in Neuronal Locking?". Neural Computation. 8 (8): 1653–1676. doi:10.1162/neco.1996.8.8.1653. ISSN 0899-7667. PMID 8888612. S2CID 1301248. <a href="https://doi.org/10.1162/neco.1996.8.8.1653" target="_blank">https://doi.org/10.1162/neco.1996.8.8.1653</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Gerstner, Wulfram. (24 July 2014). Neuronal dynamics : from single neurons to networks and models of cognition. Kistler, Werner M., 1969-, Naud, Richard., Paninski, Liam. Cambridge, United Kingdom. ISBN 978-1-107-06083-8. OCLC 861774542.{{cite book}}: CS1 maint: location missing publisher (link) <a href="978-1-107-06083-8" target="_blank">978-1-107-06083-8</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Gerstner, Wulfram. (2002). Spiking neuron models : single neurons, populations, plasticity. Kistler, Werner M., 1969-. Cambridge, U.K.: Cambridge University Press. ISBN 0-511-07817-X. OCLC 57417395. <a href="0-511-07817-X" target="_blank">0-511-07817-X</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Gerstner, Wulfram. (24 July 2014). Neuronal dynamics : from single neurons to networks and models of cognition. Kistler, Werner M., 1969-, Naud, Richard., Paninski, Liam. Cambridge, United Kingdom. ISBN 978-1-107-06083-8. OCLC 861774542.{{cite book}}: CS1 maint: location missing publisher (link) <a href="978-1-107-06083-8" target="_blank">978-1-107-06083-8</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Gerstner, Wulfram. (2002). Spiking neuron models : single neurons, populations, plasticity. Kistler, Werner M., 1969-. Cambridge, U.K.: Cambridge University Press. ISBN 0-511-07817-X. OCLC 57417395. <a href="0-511-07817-X" target="_blank">0-511-07817-X</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Gerstner, Wulfram; Hemmen, J. Leo van (1992-01-01). "Associative memory in a network of 'spiking' neurons". Network: Computation in Neural Systems. 3 (2): 139–164. doi:10.1088/0954-898X_3_2_004. ISSN 0954-898X. <a href="https://doi.org/10.1088/0954-898X_3_2_004" target="_blank">https://doi.org/10.1088/0954-898X_3_2_004</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Gerstner, Wulfram. (24 July 2014). Neuronal dynamics : from single neurons to networks and models of cognition. Kistler, Werner M., 1969-, Naud, Richard., Paninski, Liam. Cambridge, United Kingdom. ISBN 978-1-107-06083-8. OCLC 861774542.{{cite book}}: CS1 maint: location missing publisher (link) <a href="978-1-107-06083-8" target="_blank">978-1-107-06083-8</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Gerstner, Wulfram; Hemmen, J. Leo van (1992-01-01). "Associative memory in a network of 'spiking' neurons". Network: Computation in Neural Systems. 3 (2): 139–164. doi:10.1088/0954-898X_3_2_004. ISSN 0954-898X. <a href="https://doi.org/10.1088/0954-898X_3_2_004" target="_blank">https://doi.org/10.1088/0954-898X_3_2_004</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Gerstner, Wulfram. (24 July 2014). Neuronal dynamics : from single neurons to networks and models of cognition. Kistler, Werner M., 1969-, Naud, Richard., Paninski, Liam. Cambridge, United Kingdom. ISBN 978-1-107-06083-8. OCLC 861774542.{{cite book}}: CS1 maint: location missing publisher (link) <a href="978-1-107-06083-8" target="_blank">978-1-107-06083-8</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Jolivet, Renaud; Rauch, Alexander; Lüscher, Hans-Rudolf; Gerstner, Wulfram (August 2006). "Predicting spike timing of neocortical pyramidal neurons by simple threshold models". Journal of Computational Neuroscience. 21 (1): 35–49. doi:10.1007/s10827-006-7074-5. ISSN 0929-5313. PMID 16633938. S2CID 8911457. <a href="http://link.springer.com/10.1007/s10827-006-7074-5" target="_blank">http://link.springer.com/10.1007/s10827-006-7074-5</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Gerstner, Wulfram. (24 July 2014). Neuronal dynamics : from single neurons to networks and models of cognition. Kistler, Werner M., 1969-, Naud, Richard., Paninski, Liam. Cambridge, United Kingdom. ISBN 978-1-107-06083-8. OCLC 861774542.{{cite book}}: CS1 maint: location missing publisher (link) <a href="978-1-107-06083-8" target="_blank">978-1-107-06083-8</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Gerstner, Wulfram. (2002). Spiking neuron models : single neurons, populations, plasticity. Kistler, Werner M., 1969-. Cambridge, U.K.: Cambridge University Press. ISBN 0-511-07817-X. OCLC 57417395. <a href="0-511-07817-X" target="_blank">0-511-07817-X</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Gerstner, Wulfram; Hemmen, J. Leo van (1992-01-01). "Associative memory in a network of 'spiking' neurons". Network: Computation in Neural Systems. 3 (2): 139–164. doi:10.1088/0954-898X_3_2_004. ISSN 0954-898X. <a href="https://doi.org/10.1088/0954-898X_3_2_004" target="_blank">https://doi.org/10.1088/0954-898X_3_2_004</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Jolivet, Renaud; Rauch, Alexander; Lüscher, Hans-Rudolf; Gerstner, Wulfram (August 2006). "Predicting spike timing of neocortical pyramidal neurons by simple threshold models". Journal of Computational Neuroscience. 21 (1): 35–49. doi:10.1007/s10827-006-7074-5. ISSN 0929-5313. PMID 16633938. S2CID 8911457. <a href="http://link.springer.com/10.1007/s10827-006-7074-5" target="_blank">http://link.springer.com/10.1007/s10827-006-7074-5</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Gerstner, Wulfram. (24 July 2014). Neuronal dynamics : from single neurons to networks and models of cognition. Kistler, Werner M., 1969-, Naud, Richard., Paninski, Liam. Cambridge, United Kingdom. ISBN 978-1-107-06083-8. OCLC 861774542.{{cite book}}: CS1 maint: location missing publisher (link) <a href="978-1-107-06083-8" target="_blank">978-1-107-06083-8</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Gerstner, Wulfram; Hemmen, J. Leo van (1992-01-01). "Associative memory in a network of 'spiking' neurons". Network: Computation in Neural Systems. 3 (2): 139–164. doi:10.1088/0954-898X_3_2_004. ISSN 0954-898X. <a href="https://doi.org/10.1088/0954-898X_3_2_004" target="_blank">https://doi.org/10.1088/0954-898X_3_2_004</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Gerstner, Wulfram; Hemmen, J. Leo van (1992-01-01). "Associative memory in a network of 'spiking' neurons". Network: Computation in Neural Systems. 3 (2): 139–164. doi:10.1088/0954-898X_3_2_004. ISSN 0954-898X. <a href="https://doi.org/10.1088/0954-898X_3_2_004" target="_blank">https://doi.org/10.1088/0954-898X_3_2_004</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Gerstner, Wulfram (2000-01-01). "Population Dynamics of Spiking Neurons: Fast Transients, Asynchronous States, and Locking". Neural Computation. 12 (1): 43–89. doi:10.1162/089976600300015899. ISSN 0899-7667. PMID 10636933. S2CID 7832768. <a href="https://doi.org/10.1162/089976600300015899" target="_blank">https://doi.org/10.1162/089976600300015899</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">GERSTNER, Wulfram (1991). "Associative memory in a network of biological neurons" (PDF). Advances in Neural Information Processing Systems. 3: 84–90. <a href="https://papers.nips.cc/paper/371-associative-memory-in-a-network-of-biological-neurons.pdf" target="_blank">https://papers.nips.cc/paper/371-associative-memory-in-a-network-of-biological-neurons.pdf</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Gerstner, Wulfram. (24 July 2014). Neuronal dynamics : from single neurons to networks and models of cognition. Kistler, Werner M., 1969-, Naud, Richard., Paninski, Liam. Cambridge, United Kingdom. ISBN 978-1-107-06083-8. OCLC 861774542.{{cite book}}: CS1 maint: location missing publisher (link) <a href="978-1-107-06083-8" target="_blank">978-1-107-06083-8</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Gerstner, Wulfram; Hemmen, J. Leo van (1992-01-01). "Associative memory in a network of 'spiking' neurons". Network: Computation in Neural Systems. 3 (2): 139–164. doi:10.1088/0954-898X_3_2_004. ISSN 0954-898X. <a href="https://doi.org/10.1088/0954-898X_3_2_004" target="_blank">https://doi.org/10.1088/0954-898X_3_2_004</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Gerstner, Wulfram; Hemmen, J. Leo van (1992-01-01). "Associative memory in a network of 'spiking' neurons". Network: Computation in Neural Systems. 3 (2): 139–164. doi:10.1088/0954-898X_3_2_004. ISSN 0954-898X. <a href="https://doi.org/10.1088/0954-898X_3_2_004" target="_blank">https://doi.org/10.1088/0954-898X_3_2_004</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Pulsed neural networks. Maass, Wolfgang, 1949 August 21-, Bishop, Christopher M. Cambridge, Mass.: MIT Press. 1999. ISBN 978-0-262-27876-8. OCLC 42856203.{{cite book}}: CS1 maint: others (link) <a href="978-0-262-27876-8" target="_blank">978-0-262-27876-8</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">Bohte, Sander M.; Kok, Joost N.; La Poutré, Han (2002-10-01). "Error-backpropagation in temporally encoded networks of spiking neurons". Neurocomputing. 48 (1): 17–37. doi:10.1016/S0925-2312(01)00658-0. ISSN 0925-2312. <a href="http://www.sciencedirect.com/science/article/pii/S0925231201006580" target="_blank">http://www.sciencedirect.com/science/article/pii/S0925231201006580</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">Maass, Wolfgang (1997-02-01). "Fast Sigmoidal Networks via Spiking Neurons". Neural Computation. 9 (2): 279–304. doi:10.1162/neco.1997.9.2.279. ISSN 0899-7667. PMID 9117904. S2CID 34984298. <a href="https://doi.org/10.1162/neco.1997.9.2.279" target="_blank">https://doi.org/10.1162/neco.1997.9.2.279</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">Jolivet, Renaud; Rauch, Alexander; Lüscher, Hans-Rudolf; Gerstner, Wulfram (August 2006). "Predicting spike timing of neocortical pyramidal neurons by simple threshold models". Journal of Computational Neuroscience. 21 (1): 35–49. doi:10.1007/s10827-006-7074-5. ISSN 0929-5313. PMID 16633938. S2CID 8911457. <a href="http://link.springer.com/10.1007/s10827-006-7074-5" target="_blank">http://link.springer.com/10.1007/s10827-006-7074-5</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">Pozzorini, Christian; Naud, Richard; Mensi, Skander; Gerstner, Wulfram (July 2013). "Temporal whitening by power-law adaptation in neocortical neurons". Nature Neuroscience. 16 (7): 942–948. doi:10.1038/nn.3431. ISSN 1546-1726. PMID 23749146. S2CID 1873019. <a href="https://www.nature.com/articles/nn.3431" target="_blank">https://www.nature.com/articles/nn.3431</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">Mensi, Skander; Naud, Richard; Pozzorini, Christian; Avermann, Michael; Petersen, Carl C. H.; Gerstner, Wulfram (2011-12-07). "Parameter extraction and classification of three cortical neuron types reveals two distinct adaptation mechanisms". Journal of Neurophysiology. 107 (6): 1756–1775. doi:10.1152/jn.00408.2011. ISSN 0022-3077. PMID 22157113. <a href="https://journals.physiology.org/doi/full/10.1152/jn.00408.2011" target="_blank">https://journals.physiology.org/doi/full/10.1152/jn.00408.2011</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></li>
<li id="fn:38">Haslinger, Robert; Pipa, Gordon; Lima, Bruss; Singer, Wolf; Brown, Emery N.; Neuenschwander, Sergio (2012-07-03). "Context Matters: The Illusive Simplicity of Macaque V1 Receptive Fields". PLOS ONE. 7 (7): e39699. Bibcode:2012PLoSO...739699H. doi:10.1371/journal.pone.0039699. ISSN 1932-6203. PMC 3389039. PMID 22802940. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3389039" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3389039</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></li>
<li id="fn:39">Truccolo, Wilson; Eden, Uri T.; Fellows, Matthew R.; Donoghue, John P.; Brown, Emery N. (2005-02-01). "A Point Process Framework for Relating Neural Spiking Activity to Spiking History, Neural Ensemble, and Extrinsic Covariate Effects". Journal of Neurophysiology. 93 (2): 1074–1089. doi:10.1152/jn.00697.2004. ISSN 0022-3077. PMID 15356183. <a href="https://journals.physiology.org/doi/full/10.1152/jn.00697.2004" target="_blank">https://journals.physiology.org/doi/full/10.1152/jn.00697.2004</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">Pillow, Jonathan W.; Shlens, Jonathon; Paninski, Liam; Sher, Alexander; Litke, Alan M.; Chichilnisky, E. J.; Simoncelli, Eero P. (August 2008). "Spatio-temporal correlations and visual signalling in a complete neuronal population". Nature. 454 (7207): 995–999. Bibcode:2008Natur.454..995P. doi:10.1038/nature07140. ISSN 1476-4687. PMC 2684455. PMID 18650810. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2684455" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2684455</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
<li id="fn:41">Pozzorini, Christian; Mensi, Skander; Hagens, Olivier; Naud, Richard; Koch, Christof; Gerstner, Wulfram (2015-06-17). "Automated High-Throughput Characterization of Single Neurons by Means of Simplified Spiking Models". PLOS Computational Biology. 11 (6): e1004275. Bibcode:2015PLSCB..11E4275P. doi:10.1371/journal.pcbi.1004275. ISSN 1553-7358. PMC 4470831. PMID 26083597. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4470831" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4470831</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></li>
<li id="fn:42">Gerstner, Wulfram; Hemmen, J. Leo van (1992-01-01). "Associative memory in a network of 'spiking' neurons". Network: Computation in Neural Systems. 3 (2): 139–164. doi:10.1088/0954-898X_3_2_004. ISSN 0954-898X. <a href="https://doi.org/10.1088/0954-898X_3_2_004" target="_blank">https://doi.org/10.1088/0954-898X_3_2_004</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></li>
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<li id="fn:90">Mensi, Skander; Naud, Richard; Pozzorini, Christian; Avermann, Michael; Petersen, Carl C. H.; Gerstner, Wulfram (2011-12-07). "Parameter extraction and classification of three cortical neuron types reveals two distinct adaptation mechanisms". Journal of Neurophysiology. 107 (6): 1756–1775. doi:10.1152/jn.00408.2011. ISSN 0022-3077. PMID 22157113. <a href="https://journals.physiology.org/doi/full/10.1152/jn.00408.2011" target="_blank">https://journals.physiology.org/doi/full/10.1152/jn.00408.2011</a> <a href="#fnref:90" class="footnote-back-ref">↩</a></li>
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<li id="fn:94">Gerstner, Wulfram. (24 July 2014). Neuronal dynamics : from single neurons to networks and models of cognition. Kistler, Werner M., 1969-, Naud, Richard., Paninski, Liam. Cambridge, United Kingdom. ISBN 978-1-107-06083-8. OCLC 861774542.{{cite book}}: CS1 maint: location missing publisher (link) <a href="978-1-107-06083-8" target="_blank">978-1-107-06083-8</a> <a href="#fnref:94" class="footnote-back-ref">↩</a></li>
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</ol>

Spike response model open-in-new

Spike response model