| This thesis is on the application of stochastic process in computational neuroscience, including the identification of the Q matrix of ion channels with Markov chain model (the second part of the thesis) and the learning of Biological Neural Networks (the third part of the thesis), which are the newest theory and application research on nervous system cell and molecule, and computational network respectively.On the one hand, ion channels, special types of proteins embedded in membrane of the cell, including the neurons and the muscle cell, arc a kind of channel which is almost impermeable to water and water soluble molecules, controlling the movement of ions across the membrane. In certain conformation, they form a selective pore (in opening state), and allow one or more types of ions go through the pore under the voltage difference within and outside of the cell membrane, forming ion current, which makes the excitable membrane generate special electric potential change, which is the basis of the activity of nerve and muscle. But if the conformation state cannot form a pore, the channel will be in closing state.The opening and closing of ion channels is called gating. All electrical activity in the nervous system appeared to be regulated by ion channel gating. The development of dynamic model of gating mechanisms and research of relative problems is the core subject in the fields of ion channel. Colquhoun and so on [15-17, 42, 44, 45] has made outstanding contribution in the stochastic process theory of ion channels, and developed the Markov model of ion channel gating kinetics, which is accepted by most researchers. Only few states (opening and closing) is observable. These states are called open states. As a result, the research of gating kinetics of ion channels is mainly focused on determining all the underlying transition rates of Markov chain via observation and statistics of the few opening states. The transition rate of Markov chain is the Q Matrix we often refer to, which is called gating kinetics or parameters of model.Currently, as for the special mechanisms of ion channels, maximum likelihood method is directy adopted for the estimate of parameters of model. Although this method is powerful, according to the research of the past few years, there are still some defects in the following aspects: 1, the transition rates estimated by likelihood function is not exclusive, because the same likelihood fhnction can be generated by different Q matrix, therefore the number of states of Markov chain are constrained (Even if the constraint is not violated, it can not be fixed, see Section 3.7 in details); 2, the quantity of calculation is too great, the time of computation is too long, and it is not so accurate; 3, for most of the users, it is difficult to adopt the measurements of variables by searching the corresponding likelihood curved surface.Because of the above defects, with the use of the intrinsic properties of Markov chain and matrix analytical method, the thesis attempts to find out the constrained equations between the opening states of life-time and death-time and transition rates. The method with which the transition rates are fixed according to the necessary constrained equations and their recursive relation is named Markov Chain Back-calculation Method. According to the achievements of the research in the thesis, this method is of a variety of advantages: 1, with the integration of the characteristic of Markov chain model, the transition rate achieved is exclusive, the states are unlimited; 2, little numeration, short time in calculation, accuracy in accounting; 3, the algorithm, being exact, can be programmed, easy to be adopted.On the other hand, the nervous system is the complex Biological Neural Networks, BNN, constructed by numerous neurons, with the function of learning and memory, making the human talented. Artificial Neural Networks, ANN, as the simple and rather similar form of BNN, and its learning theory, have evolved unprecedentedly. And its wide application in other fields has been carried out. With the construction of the BNN which is more and more biologically, especially the development of the theory of the Moment Neural Networks [81], which encoding the first and second order statistics of the neuronal outputs. Naturally, what would be asked is whether perceptrons theory, which might be more close to the reality of creatures, could be constructed with the introduction of the learning theory of ANN, also called classical perceptrons into BNN. And are they of the characteristics that classical ones don't own? Can they perform the more complicated tasks?Therefore, some of the principles of ANN have been introduced into BNN so as to construct three kinds of perceptrons, which may be more and more close to the reality of the creature, the first, second and generalized second-order spiking perceptrons and their application is explored in a brief way. It is testified in the thesis that they are actually of the characteristics that are not owned in classical ones and they can perform more difficult tasks.The conclusions about the research in the two aspects and achievements of the thesis are as follows:1. Based on the intrinsic properties of Markov chain, with the use of the nature of symmetrical transition function, it is proved that probability density function, PDF, of life-time or death-time of a certain state or a set of states, is (mixture) exponential density. And constrained equations between derivatives, moments of PDF at 0 moment and exponential of PDF and transition rates of Markov chain have been carried out. (see Section 3.1 and Section 4.2.5)2. According to the characteristics of undering Markov chain in different ion channels, with the use of necessary constrained equations Markov chain Back-calculation Method, which is applied to determine its transition rates, has been put forward. The main ideas are in the following: firstly, the relevant PDF has been fixed according to few lifetime and death-time series under opening states. Then the necessary constrained equations and recursive relation have been found out according to the undering characteristics of Markov chain. And lastly, the whole transition rates (Q matrix) have been worked out.a). The identification of transition rates of some fundamental and common Markov models have been demonstrated. According to their different characteristics, the specific algorithm for transition rates has been given respectively under different conditions of observation. Related numeric examples are given so as to approve the validity of the conclusion and the applicability to gating kinetics of ion channels. The models include the following forms: cyclic, linear, star-graph, star-graph branch, hierarchical Markov chain and the Markov chain with a loop. (see Chapter 3)b) For some common Markov models, the idea to observe their sub-models, such as linear and cyclic sub-models, has been proposed, and identifiable conditions and related algorithm are given. Some direct and general conclusions and principles are concluded. (See chapter 4)c). The evaluative method about identifying of the undering Markov chain structure in ion channels and propagation of errors have been explored, showing that transition rates determined by Markov Chain Back-calculation Method will not enlarge the error aroused by the fitting of the mixture exponential distribution.3. The advantages of Markov Chain Back-calculation Method are as follows: (1), the transition rate achieved is exclusive, the states are unlimited; (2), little numeration, short time in calculation, accuracy in accounting (only if the related PDF is accurate); 3, the algorithm, being exact, can be programmed, easy to be adopted.Especially, this method can solve some problems which can't be done with the maximum likelihood Method. For example, as for the most simple cyclic ion channels stated in [195], on any occasion, Markov Chain Back-calculation Method can confirm the transition rates. (See Section 3.7 and Section 4.2.5.3)4. Therefore, some of the principles of ANN havc been introduced into BNN so as to develop three kinds of perceptrons, which may be more and more close to the reality of the creature, the first, second and generalized second-order spiking perceptrons.5. The three kinds of perceptrons are actually of the characteristics that are not owned in classical ones and they can perform more difficult non-linear tasks.a). All of the single layered spiking perceptrons can be trained to perform nonlinear tasks exclusively and to successfully classify the XOR problems, which can not be achieved by the traditional single layered perceptrons.b). (Generalized) second-order spiking perceptrons can train not only the mean of the output, but also the variance. And the errors derived from the mean and variance of output can be trade-off according to the practical application.c). Second-order spiking perceptrons can perform any complicated non-linear tasks, and classical learning tasks—simulating the trajectory of the movement of the arm, it is therefore shown that this can also be applied in another classical learning task—function approximation.It is mentioned that all of the simulation and computation in the thesis are completed in Matlab7.0, and its related programs are compiled by myself. Especially for those programs about the complicated simulation and numerical calculation in learning of Biological Neural Networks, a lot of time and energy have been spent.Besides, the author has also done some work on the following fields such as dynamics of (generalized) second-order spiking networks, neuronal decoding and so on. But because of the system of the whole thesis, they have not been introduced.
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