Statistical Methods In Neuronal Dynamical Data

There are approximately 10¹¹neurons forming high dimensional neuronal networks in a human brain.In neuroscience,it is an important problem of how to deal with high dimensional neuronal data.In this thesis,we develop effective methods to deal with high dimensional spike-trains obtained from neuronal networks.The structure of data can be regarded by correlations of all orders.We study the relation among different orders of data structure under Maximum Entropy?MaxEnt?principle,where the MaxEnt principle is widely used in diverse fields.It has been reported that by using only observed firing rates and second-order correlations,the second-order MaxEnt model can well approximate the full distribution of neuronal firing patterns in many neuronal networks.Thus,it confers great advantage of the second-order MaxEnt model in that the structure of correlations in the analysis of neuronal activity data reduces drastically to only two orders.We address the issue of why the second-order MaxEnt model can successfully describe the distribution of neuronal firing patterns.I have three contributions to this issue:First,we found a widely used index for the performance of the second-order MaxEnt model is misleading,which gives an example to show the erroneous result by using entropy indiscreetly;Second,via the perturbative analysis,we explore a possible dynamical state in which the second-order MaxEnt model can be a good effective description;Third,our theoretical framework is helpful for the experimentalists to deal with data obtained from real neurons.It is shown from the study of the MaxEnt model that the correlations among neurons is very important,whereas it lacks of methods that can incorporating with correlations to deal with neuronal data obtained in short recordings?hundreds of milliseconds?.We develop a method to pretreat neuronal data,which transforms the correlation of every individual neuron in time to the spatial correlations between neurons.Based on this pretreatment,we can perform high dimensional two-sample test to see the coding difference of different stimuli in neuronal net-works.Another contribution of this thesis is that the developed method is able to deal with high dimensional neuronal data in short recordings.With the development of new experimental techniques that can measure multi-neuron signals,many laboratories have been able to perform long-duration recordings from hundreds of neurons simultaneously,therefore,the developed method is very useful for experimentalists.