1. Ch 3. Likelihood Based Approach to Modeling the Neural Code Bayesian Brain: Probabilistic Approaches to Neural Coding eds. K Doya, S Ishii, A Pouget, and R Rao Dec. 18 th, 2008 Summarized by Seok Ho-Sik © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/1

2. Task, Conclusion, Contribution Task: design a model depicting the probabilistic relationship between stimuli and neural response. Conclusion  ML method is useful.  ML  Problem:  in a high-dimensional space, containing tens to hundreds of parameters(describing a neuron’s receptive field and spike-generation properties) Model Contributions  Introducing neural coding models, their validating methods(likelihood- based cross-validation, time-rescaling, model-based decoding).  For us, introducing what should be considered to model neural responses (nonlinearity, probabilistic relation etc.). © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 2 ( ) LNP GLM Generalized Integrate-and-Fire Model

3. 3.1 The Neural Coding Problem(1/2) © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 3

4. 3.1 The Neural Coding Problem(2/2) Probabilistic relations: relationship between stimuli and neural response is probabilistic. Difficulties for obtaining full response distributions  The high dimensionality of stimulus space and the finite duration of neurophysiology experiments. A classical approach Assumption: neurons are sensitive to a restricted set of stimulus features. Statistical approach: assumes a probabilistic model of neural response and attempts to fit the model parameter   Goal: to find a simplified and computationally tractable description of p(y|x). © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 4

5. 3.2 Model Fitting with Maximum Likelihood Data: Given a particular model, parameterized by the vector , one can apply a maximum likelihood method to obtain an asymptotically optimal estimate of  Difficulties  For many models of neural response (e.g. detailed biophysical models) it is very difficult to compute likelihood.  In most cases,  lives in a high-dimensional space. © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/

6. 3.2.1 The LNP Model (1/5) The LNP (linear-nonlinear Poisson) model. The cascading of a linear filter(k), a point nonlinearity (f), and Posisson spike generator.  k: represents the neuron’s space-time receptive field, which describes how the stimulus is converted to intracellular voltage.  f: the conversion of voltage to an instantaneous spike rate.  Instantaneous rate is converted to a spike train via an Poisson process. Model parameters: : the parameters governing f. © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/

7. 3.2.1 The LNP Model (2/5) Because the bins of the responses are conditionally independent of one another given the stimulus, The second term on the right hand of the (  ) converges to a vector proportional to k if the stimulus distribution p(x) is spherically symmetric. The first term is proportional to the STA (spike-triggered average) if f`/f is constant (when f is exponential). y i : the spike count in the i th time bin, x i : the stimulus vector associated with this bin. Rate parameter,Dot product,Δ: the width of the time bin …(  ) © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ Differentiating with respect to K

8. 3.2.1 The LNP Model (3/5) A comparison between the STA and the ML estimates of the linear filter k on spike trains simulated using three different nonlinearities. Result  The ML estimate outperforms the STA except when f is exponential.  If f differs significantly from exponential, the traditional STA is suboptimal. © 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/

9. 3.2.1 The LNP Model (4/5) A comparing ML to an estimator derived from STC (spike-triggered covariance), which uses the principal eigenvector of the STC matrix to estimate k. Result: the ML estimate outperforms the STC and STA except when f is exponential.

10. 3.2.1 The LNP Model (5/5) Limits  The LNP model is not biophysically realistic (especially the assumption of Poisson spiking).  Computationally intensive.  Can not be guaranteed to converge to global maximum. Usefulness  It provides a compact and reasonably accurate description of average response in many early sensory areas.  LNP model can be generalized to include multiple linear filters and a multidimensional nonlinearity.  LNP models could use information-theoretic estimators for finding “maximally informative dimensions” or features of stimulus space.  Because its sensitivity to higher-order statistics of the spike-triggered ensemble, it is more powerful and more general than STA or STC.

11. 3.2.2 Generalized Linear Model (1/2) Incorporating feedback from the spiking process, allowing the model to account for history-dependent properties of neural spike trains. If we let denote the instantaneous spike rate as following, then

12. 3.2.2 Generalized Linear Model (2/2) How to discover the global maximum?  To constrain the model so that one can guarantee that the likelihood function is free from local maxima.  If one can show that the negative log-likelihood is convex, then the only maxima is the global maxima.  The problem of computing the ML estimate is reduced to a convex optimization problem, for which there are tractable algorithms. A number of suitable functions seems like reasonable choices for describing the conversion of intracellular voltage to instantaneous spike rate (e. g.). The GLM framework is quite general, and can easily be expanded to include additional linear filters that capture dependence on spiking activity in nearby neurons, behavior of the organism, or additional external covariates of spiking activity.

13. 3.2.3 Generalized Integrate-and-Fire Model (1/3) Recent work: the leaky integrate-and-fire (IF) model, a canonical but simplified description of intracellular spiking dynamics, can reproduce the spiking statistics of real neurons and can mimic important dynamical behaviors of more complicated models. The injected current: a linear function of the stimulus, the spike-train history, plus a Gaussian noise current that introduces a probability distribution over voltage trajectories.

14. 3.2.3 Generalized Integrate-and-Fire Model (2/3) The model of dynamics The dependency structure: the probability of an entire interspike interval is depending on a relevant portion of the stimulus and spike-train history.

15. 3.2.3 Generalized Integrate-and-Fire Model (3/3) How to compute the likelihood for a single ISI under the generalized GIF model using Monte Carlo sampling  Given a setting of the model parameters, one can sample voltage trajectories from the model, drawing independent noise samples for each trajectory, and following each trajectory until it hits threshold.  The probability of a spike occurring at the ith bin is simply the fraction of voltage paths crossing threshold at this bin. Likelihood function Sample paths Voltage path obtained in the absence of noise

16. 3.3 Model Validation (1/2) Validating the quality of the model fit. Likelihood-Based Cross-validation  Time Rescaling  To use the model to convert spike times into a series of i.i.d random variables.  Given the cumulative density function for a random variable, a general result from probability theory holds that it provides a remapping of that variable to the one randomly distributed unit interval[0,1].  Any correlation (or some other form of dependence) between successive pairs of remapped spike times, indicates a failure of the model.

17. 3.3 Model Validation (2/2) Model-Based decoding  To perform stimulus decoding using the model-based likelihood function.  One can obtain the most likely stimulus to have generated the response y by maximizing the posterior for x, which gives the maximum a posteriori estimate of the stimulus  With mean-squared error, this estimator is given by  Decoding allows to measure how well a particular model preserves the stimulus-related information in the neural response.  Even though a model fails to reproduce certain statistical features of the response, it provides a valuable tool for assessing what information the spike train carries about the stimulus and gives a perhaps more valuable description of the neural code.