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<h1 class="title">Decoding of feature selectivity in neural activity</h1>
<h1>Concrete applications in visual data</h1>
<h2>Laurent U Perrinet, INT - Wahiba Taouali, INMED</h2>
<img src="figures/troislogos.png" width=61%/>
""")
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Decoding of feature selectivity in neural activity

Concrete applications in visual data

Laurent U Perrinet, INT - Wahiba Taouali, INMED

  • Course on Computational Neuroscience, Marseille, December 8th, 2015

Acknowledgements:

  • PhD program: Anna Montagnini, Frédéric Chavane, Nadia Pittet, INT, Marseille
  • Material: Peggy Series, Christopher Olah
  • NeuroPhysiology: Giacomo Benvenuti, Frédéric Chavane, INT, Marseille

ⓦ https://laurentperrinet.github.io/sciblog/files/2015-12-08_cours_neurocomp/PerrinetEtAl12neurocomp-intro.slides.html

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Outline

Outline

Bayes : travelling back to the feature space

(see http://colah.github.io/posts/2015-09-Visual-Information/)

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examples of Bayesian mechanism in perception

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# http://jov.arvojournals.org/article.aspx?articleid=2121565
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summary

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summary

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Outline

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<h2>Optimal representation of sensory information </h2>
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<tr> 
<td width=50%/><img src="{0}" width=75%/> </td>
<td width=50%/><img src="{1}" width=100%/> </td>
</tr> 
</table>
<em>Mehrdad Jazayeri & J Anthony Movshon</em> (2007) Nature Neuroscience
""".format(figpath + 'Jazayeri07optimal_figure1.png', figpath + 'Jazayeri07optimal_figure2.png'))
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Optimal representation of sensory information

Mehrdad Jazayeri & J Anthony Movshon (2007) Nature Neuroscience

let's look at the proof

  • The definition of a tuning function that accounts for the modulation of the parameters of the variability model (mean, shape, scale, std) when we vary the stimili : f($\theta$)= mean(k|$\theta$) which gives : \begin{equation} P(k|{\theta}) = \frac{f({\theta}) ^{k}e^{-f({\theta})}}{k!} \end{equation}

  • The pooling of the population information : by the independence hypothesis, the probability of having a population response Y = [k{1}, k{2}.. k_{N}] (vector of N cells responses):

\begin{equation} P(Y|\theta) = \Pi ^N _{i=1} P(k_{i}|\theta) \end{equation}
  • Bayes' rule.
\begin{equation} P({\theta}|Y)=\frac{ P(Y|{\theta}) P(\theta) }{P(Y)} \end{equation}

Maximum likelihood decoding.

The decoding algorithm consists of maximizing the posterior probability $P({\theta}|Y)$ as a function of the estimated direction ${\theta}$, given a distribution hypothesis:

  • The evidence term $P(Y)$ is a normalization term independent of ${\theta}$ $\to P(Y)$=cst
  • There is no prior knowledge on ${\theta}$ (such that $ \forall \theta_1, \theta_2$, $P(\theta_1)$ = $P(\theta_2)$)

Thus, maximizing the posterior $P(\theta|Y)$ under the Poisson hypothesis is equivalent to maximizing the following likelihood function: \begin{equation} L(\theta) = P(Y|{\theta}) = \Pi ^N _{i=1} \frac{f_i({\theta}) ^{k_i}e^{-f_i({\theta})}}{k_i!} \end{equation}

In practice, It is often the log-likelihood function that is considered:

\begin{equation} LL(\theta) = log(P(Y|{\theta})) = \sum_{i=1}^N{k_i\log[f_{i}(\theta)]}-\sum_{i=1}^N{f_{i}(\theta)}- \sum_{i=1}^N \log[{k_i!}] \end{equation}

In the end:

\begin{equation} LL(\theta) = \sum_{i=1}^N{k_i\log[f_{i}(\theta)]} - \log(Z) \end{equation}

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 <h2>Poisson distribution as a model of variability </h2>
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Poisson distribution as a model of variability

Outline

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python in neuroscience

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Python in Neuroscience

    ● Scientific python tools : Numpy, Scipy

    ● Plotting tools : Matplotlib

    ● Python interfaces to most major neuroscience software tools

        ● e.g. PyNN, PyNEURON, PyNEST, Brian

        ● Neurotools

Python vs. Matlab

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Ipython Notebook

    ● Interactive shell

    ● enhanced introspection,

    ● code highlighting

    ● tab completion
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<h1 class="title">Decoding of feature selectivity in neural activity</h1>
<h1>Concrete applications in visual data</h1>
<h2>Laurent U Perrinet, INT - Wahiba Taouali, INMED</h2>
<img src="figures/troislogos.png" width=61%/>
""")
Out[27]:

Decoding of feature selectivity in neural activity

Concrete applications in visual data

Laurent U Perrinet, INT - Wahiba Taouali, INMED

Acknowledgements:

  • PhD program: Anna Montagnini, Frédéric Chavane, Nadia Pittet, INT, Marseille
  • Material: Peggy Series, Christopher Olah
  • NeuroPhysiology: Giacomo Benvenuti, Frédéric Chavane, INT, Marseille

ⓦ https://laurentperrinet.github.io/sciblog/files/2015-12-08_cours_neurocomp/PerrinetEtAl12neurocomp-intro.slides.html