Linear-nonlinear-Poisson cascade model

<h2 id="mathematical-formulation">Mathematical formulation</h2>
<h3>Single-filter LNP</h3>
<p>Let 
  
    
      
        
          x
        
      
    
    {\displaystyle \mathbf {x} }
  
 denote the spatio-temporal stimulus vector at a particular instant, and

k
        
      
    
    {\displaystyle \mathbf {k} }
  
 denote a linear filter (the neuron's linear receptive field), which is a vector with the same number of elements as 
  
    
      
        
          x
        
      
    
    {\displaystyle \mathbf {x} }
  
.  Let 
  
    
      
        f
      
    
    {\displaystyle f}
  
 denote the nonlinearity, a scalar function with non-negative output.  Then the LNP model specifies that, in the limit of small time bins, 
</p>

P
        (
        
          
            spike
          
        
        )
        ∝
        f
        (
        
          k
        
        ⋅
        
          x
        
        )
      
    
    {\displaystyle P({\textrm {spike}})\propto f(\mathbf {k} \cdot \mathbf {x} )}
  
.
<p>For finite-sized time bins, this can be stated precisely as the probability of observing <i>y</i> spikes in a single bin: 
</p>

P
        (
        y
        
          
            ~spikes
          
        
        )
        =
        
          
            
              
                (
                
                  Δ
                  λ
                
                )
              
              
                y
              
            
            
              y
              !
            
          
        
        
          e
          
            −
            Δ
            λ
          
        
      
    
    {\displaystyle P(y{\textrm {~spikes}})={\frac {\left(\Delta \lambda \right)^{y}}{y!}}e^{-\Delta \lambda }}

where 
  
    
      
        λ
        =
        f
        (
        
          k
        
        ⋅
        
          x
        
        )
      
    
    {\displaystyle \lambda =f(\mathbf {k} \cdot \mathbf {x} )}
  
, and 
  
    
      
        Δ
      
    
    {\displaystyle \Delta }
  
 is the bin size.
<h3>Multi-filter LNP</h3>
<p>For neurons sensitive to multiple dimensions of the stimulus space, the linear stage of the LNP model can be generalized to a bank of linear filters, and the nonlinearity becomes a function of multiple inputs.  Let 
  
    
      
        
          
            k
            
              1
            
          
        
        ,
        
          
            k
            
              2
            
          
        
        ,
        …
        ,
        
          
            k
            
              n
            
          
        
      
    
    {\displaystyle \mathbf {k_{1}} ,\mathbf {k_{2}} ,\ldots ,\mathbf {k_{n}} }
  
 denote the set of linear filters that capture a neuron's stimulus dependence.  Then the multi-filter LNP model is described by
</p>

P
        (
        
          
            spike
          
        
        )
        ∝
        f
        (
        
          
            k
            
              1
            
          
        
        
        ⋅
        
        
          x
        
        ,
        
        
          
            k
            
              2
            
          
        
        
        ⋅
        
        
          x
        
        ,
        
        …
        ,
        
        
          
            k
            
              n
            
          
        
        
        ⋅
        
        
          x
        
        )
      
    
    {\displaystyle P({\textrm {spike}})\propto f(\mathbf {k_{1}} \!\cdot \!\mathbf {x} ,\;\mathbf {k_{2}} \!\cdot \!\mathbf {x} ,\;\ldots ,\;\mathbf {k_{n}} \!\cdot \!\mathbf {x} )}

P
        (
        
          
            spike
          
        
        )
        ∝
        f
        (
        K
        
          x
        
        )
        ,
      
    
    {\displaystyle P({\textrm {spike}})\propto f(K\mathbf {x} ),}

<p>where 
  
    
      
        K
      
    
    {\displaystyle K}
  
 is a matrix whose columns are the filters 
  
    
      
        
          
            k
            
              i
            
          
        
      
    
    {\displaystyle \mathbf {k_{i}} }
  
.
</p>
<h2 id="estimation">Estimation</h2>
<p>The parameters of the LNP model consist of the linear filters 
  
    
      
        {
        
          
            k
            
              i
            
          
        
        }
      
    
    {\displaystyle \{{k_{i}}\}}
  
 and the nonlinearity 
  
    
      
        f
      
    
    {\displaystyle f}
  
.  The estimation problem (also known as the problem of <i>neural characterization</i>) is the problem of determining these parameters from data consisting of a time-varying stimulus and the set of observed spike times. Techniques for estimating the LNP model parameters include:
</p>
<ul><li>moment-based techniques, such as the <a href="/facts/Spike-triggered_average/MNU4wMst">spike-triggered average</a> or <a href="/facts/Spike-triggered_covariance/QcLjRNXf">spike-triggered covariance</a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li>
<li>with information-maximization or <a href="/facts/Maximum_likelihood/0Yq2dpQD">maximum likelihood</a> techniques.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a></li></ul>
<h2 id="related-models">Related models</h2>
<ul><li>The LNP model provides a simplified, mathematically tractable approximation to more biophysically detailed <a href="/facts/Biological_neuron_model/eCPeWaXc">single-neuron models</a> such as the <a href="/facts/Integrate-and-fire/eCPeWaXc">integrate-and-fire</a> or <a href="/facts/Hodgkin%25E2%2580%2593Huxley_model/HMuK0gP9">Hodgkin–Huxley model</a>.</li>
<li>If the nonlinearity 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is a fixed invertible function, then the LNP model is a <a href="/facts/Generalized_linear_model/Sf1htm5W">generalized linear model</a>.  In this case, 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is the inverse link function.</li>
<li>An alternative to the LNP model for neural characterization is the <a href="/facts/Volterra_series/oDUJXYxR">Volterra kernel</a> or <a href="/facts/Wiener_kernel/8DkBwEMs">Wiener kernel</a> series expansion, which arises in classical nonlinear systems-identification theory.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>  These models approximate a neuron's input-output characteristics using a polynomial expansion analogous to the <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a>, but do not explicitly specify the spike-generation process.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Random_neural_network/LLlVYyVz">Random neural network</a></li>
<li><a href="/facts/Spike-triggered_average/MNU4wMst">Spike-triggered average</a></li>
<li><a href="/facts/Spike-triggered_covariance/QcLjRNXf">Spike-triggered covariance</a></li></ul>

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

<ol>
<li id="fn:1"><p>Chichilnisky, E. J., A simple white noise analysis of neuronal light responses. Archived 2008-10-07 at the Wayback Machine Network: Computation in Neural Systems 12:199–213. (2001) <a href="http://www.stat.columbia.edu/~liam/teaching/neurostat-fall13/papers/Chichilnisky-2001.pdf" target="_blank">http://www.stat.columbia.edu/~liam/teaching/neurostat-fall13/papers/Chichilnisky-2001.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Simoncelli, E. P., Paninski, L., Pillow, J. & Swartz, O. (2004). Characterization of Neural Responses with Stochastic Stimuli in (Ed. M. Gazzaniga) The Cognitive Neurosciences 3rd edn (pp 327–338) MIT press. <a href="http://www.cns.nyu.edu/~eero/ABSTRACTS/simoncelli03c-abstract.html" target="_blank">http://www.cns.nyu.edu/~eero/ABSTRACTS/simoncelli03c-abstract.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Schwartz O., Pillow J. W., Rust N. C., & Simoncelli E. P. (2006).  Spike-triggered neural characterization.  Journal of Vision 6:484–507 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Chichilnisky, E. J., A simple white noise analysis of neuronal light responses. Archived 2008-10-07 at the Wayback Machine Network: Computation in Neural Systems 12:199–213. (2001) <a href="http://www.stat.columbia.edu/~liam/teaching/neurostat-fall13/papers/Chichilnisky-2001.pdf" target="_blank">http://www.stat.columbia.edu/~liam/teaching/neurostat-fall13/papers/Chichilnisky-2001.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Simoncelli, E. P., Paninski, L., Pillow, J. & Swartz, O. (2004). Characterization of Neural Responses with Stochastic Stimuli in (Ed. M. Gazzaniga) The Cognitive Neurosciences 3rd edn (pp 327–338) MIT press. <a href="http://www.cns.nyu.edu/~eero/ABSTRACTS/simoncelli03c-abstract.html" target="_blank">http://www.cns.nyu.edu/~eero/ABSTRACTS/simoncelli03c-abstract.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Schwartz O., Pillow J. W., Rust N. C., & Simoncelli E. P. (2006).  Spike-triggered neural characterization.  Journal of Vision 6:484–507 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Brenner, N., Bialek, W., & de Ruyter van Steveninck, R. R. (2000). <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Paninski, L. (2004) Maximum likelihood estimation of cascade point-process neural encoding models. In Network: Computation in Neural Systems. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Mirbagheri M. (2012) Dimension reduction in regression using Gaussian Mixture Models. In Proceedings of International Conference on Acoustics, Speech and Signal Processing (ICASSP). <a href="https://ieeexplore.ieee.org/document/6288342/?arnumber=6288342&tag=1" target="_blank">https://ieeexplore.ieee.org/document/6288342/?arnumber=6288342&tag=1</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Marmarelis & Marmerelis, 1978. Analysis of Physiological Systems: The White Noise Approach. London: Plenum Press. <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
</ol>

Linear-nonlinear-Poisson cascade model open-in-new

Linear-nonlinear-Poisson cascade model