Stochastic diffusion processes with jumps for cancer growth and neuronal activity models

2013 - 2014In the last decades, great attention has been paid to the description of bio- logical, physical and engineering systems subject to various types of jumps. A jump, or catastrophe, is considered as a random event that shifts the state of an evolutionary process in a certain level from which the process can re- start. A catastrophe can represent the extinction or reduction of elements in a biological population (due to virus infection or external agent) or of customers in a queue system (due to power failure, reset or system bug). In literature, some results have been obtained for continuous-time Markov chains and stochastic diffusion processes subject to catastrophes occurring at exponential rate. In this thesis we propose to study further evolutionary processes subject to jumps and we consider various applications of interest in different areas. In particular, we introduce the effect of jumps in: • deterministic models for rumor spreading, • time non-homogeneous Markov chains, • stochastic diffusion processes with particular attention to the Gom- pertz model for cancer evolution and to the non-homogeneous Ornstein- Uhlenbeck process for neuronal activity. Specifically, we analyze firstly rumor spreading mechanisms, during which one can consider the effect of an external entity that denies the rumor so that the process is reset to the initial state consisting in a unique spreader that renews the spreading process. This study is provided in the subsection of the Introduction Rumor spreading with denials. The denials, or jumps, are random and they occur according to a Poisson process with parameter . Two rumor spreading models with denials are studied. In both models the population is divided into three groups: the spreaders (who know and transmit the rumor), the ignorants (who do not know the rumor) and the stiflers (who know the rumor but do not transmit it). The rumor spreads through pair-wise contacts, occurring with rate , between spreaders and the other people. We consider a model A based on the well-known DK where denials are intro- duced and we study an alternative model, model B, in which denials occur and each spreader can transmit the rumor at most k times. For both models, we write the system of ordinary differential equations describing the rumor spreading mechanism and we study its steady state solution focusing on the asymptotic percentage of ignorants to identify the density of the population that knows the rumor. A scrutinized numerical analysis is performed to study the effect of denials on varying parameters and to compare the pro- posed models. We note that, in both cases the asymptotic percentage of ignorants increases when the rate of the denials grows respect to the rate of the contacts; in particular, if the size of the population is large and , the rumor does not spread at all. For the model B, the density of individuals that knows the rumor increase with k, since the rumor has more chance to spread. Moreover, the model B behaves like the model A when k increases, in particular a good match is found already for k = 6. Finally, in both models we obtain that at most the half of the population can be informed about the rumor. Concerning the time non-homogeneous Markov chains, we consider a queue- ing system subject to catastrophes which occur at random times and that empty instantaneously the system reducing to zero the number of customers. This study is shown in the subsection of the Introduction Time non-homogeneous adaptive queue with catastrophes. Catastrophes occur according to a time non-homogeneous Poisson process; in particular, the catastrophe’s rates depend on time and on the number of customers in the queue. We analyze the system by studying the transition probabilities and the mo- ments of the number of customers in the system. We focus on the problem of the first visit time (FVT) to zero state with particular attention to busy pe- riod of the service center, i.e the time interval during which at least one server is busy. Specifically, we pay attention to the case in which the catastrophe intensity is a periodic function of time obtaining some properties of asymp- totic distribution and of the FVT density. We study the M/M/1 queueing systems to perform an example of the obtained results. After a brief study of deterministic models and of Markov chains subject to jumps, the thesis is focused especially on stochastic diffusion processes with jumps. In Chapter 1, Stochastic diffusion processes with random jumps, we construct diffusion processes with jumps by supposing that catastrophes occur at time interval following a general distribution and the return points are randomly chosen. Moreover, we consider the possibility that, after each jump, the process can evolve with a different dynamics respect to the previ- ous processes; we also suppose that the inter-jump intervals and the return points are not identically distributed. For this type of process, we analyze the probability density function (pdf), its moments and the first passage time (FPT) problem. We also study the Wiener process with jumps, as example. In the remaining chapters of the thesis, we focus on the effect of jumps in stochastic diffusion processes of interest in neurobiology. In Chapter 2, A Gompertz model with jumps for an intermittent treatment in cancer growth, we construct a Gompertz process with jumps to analyze the effect of a therapeutic program that provides intermittent suppression of cancer cells. In this context, a jump represents an application of the therapy. Firstly, we consider a simple model in which the Gompertz process has the same characteristics between two consecutive jumps, the return points and the inter-jump intervals are independent and identically distributed. For this model, we study the transition pdf, the average state of the system (representing the mean size of the tumor) and the number of therapeutic applications to be carried out in time intervals of fixed amplitude. We consider the degenerate and the exponential distribution for the inter-jump intervals and we study three different distributions of the return point (degenerate, uniform and bi-exponential). We note that the obtained results for different distribution are comparable, so, in the following studies, we consider only the degenerate. After this first step, we construct a more realistic model. Specifically, we as- sume: the therapeutic program has a deterministic scheduling, so that jumps occur at fixed and conveniently chosen time instants; the return points are deterministic; therapeutic treatments weaken an ill organism and when a therapy is applied there is a selection event in which only the most aggres- sive clones survive (for example this perspective could be applied to targeted drugs that have a much lower toxicity for the patient). Taking into consideration these aspects, we construct the deterministic and stochastic processes with jumps. Since each therapeutic application involves a reduction of the tumor mass, but it also implies an increase of the growth speed, the problem of finding a compromise between these two aspects raises. Two possible scheduling are proposed in order to control the cancer growth. In the first scheduling, we assume that inter-jump intervals have equal size. We also suppose that the return points are all equal after each jump. In this case, we obtain interesting properties which allow to choose the most appropriate application times, when the toxicity of the drug is fixed. In the second scheduling, we suggest to apply the therapy just before the cancer mass reaches a fixed control threshold S. To this aim, we study the FPT problem through S and we provide information on how to choose the application times so that the cancer size remains bounded during the treat- ment. The goodness of the obtained results is measured via the increase of the mean FPT of the process through S. The performed analysis shows that better results are obtained when the therapy is applied as later as possible, for higher control thresholds and smaller weakening rates. Moreover, we compare the deterministic and stochastic approaches noting that, for both scheduling, the mean FPT through S increases as the amplitude of random fluctuations increases. We also provide a comparison between the two proposed scheduling and we conclude that the second strategy is the best, i.e. it is preferable to apply the therapy just before the cancer mass crosses the control threshold. In Chapter 3, Return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model, we consider a diffusion stochastic pro- cess with jumps for the neuronal activity. To describe the input-output behavior of a single neuron subject to a diffusion- like dynamics, we model the neuronal membrane potential via the Ornstein- Uhlenbeck (OU) diffusion process. We assume that inputs, while remaining a constant amplitude, are characterized by time-dependent rates. In partic- ular, we consider an OU process characterized by a time-dependent drift in which appears a periodic function m(t) representing some oscillatory effects of the environment acting on the neuron. To describe a neuronal train spike, a return process is constructed on such time non-homogeneous OU process as follows. Starting from the value rep- resenting the resting potential, the neuronal membrane potential follows the non-homogeneous OU process as long as a threshold (the action threshold) is reached for the first time. In correspondence to the reaching of this peak, a neuronal spike occurs resetting the process to the resting potential. Then, the membrane potential evolves as before until the threshold is reached again causing another neuronal spike, and so on. In order to study the interspike intervals (ISI) distribution, we analyze the FPT random variable of the non-homogeneous OU process because it rep- resents the theoretical counterpart of the neuronal firing time, so that the FPT’s pdf describes the pdf of the firing time. In this regard, we make use of an asymptotic behavior of exponential type for the FPT pdf . Concerning this return process, we study the ISI distribution and the number of firings occurring until a fixed time. Moreover, we take into account the effect of the refractoriness on the model. A refractory period is a time interval following each spike and during which the neuron is completely or partially unable to respond to stimuli. Hence, we introduce random downtimes which delay spikes, simulating the effect of refractoriness. We provide the expression of the ISI distribution also for the process with refractoriness. This distribution is conditioned by the time in which the last fire occurs. A theoretical and numerical analysis of the return process in the presence of constant and exponential refractoriness is performed. Some similarities between the ISI pdf with refractoriness and without refrac- toriness are observed. In particular, our analysis shows that the ISI pdf in the presence of refractoriness is shifted respect to the ISI pdf in the absence of refractoriness, provided the latter is suitably conditioned. This observa- tion supports the proposed model. The thesis ends with conclusions on the obtained results and with some possible future developments. [edited by author]XIII n.s

Low-dimensional firing-rate dynamics for populations of renewal-type spiking neurons

The macroscopic dynamics of large populations of neurons can be mathematically analyzed using low-dimensional firing-rate or neural-mass models. However, these models fail to capture spike synchronization effects of stochastic spiking neurons such as the non-stationary population response to rapidly changing stimuli. Here, we derive low-dimensional firing-rate models for homogeneous populations of general renewal-type neurons, including integrate-and-fire models driven by white noise. Renewal models account for neuronal refractoriness and spike synchronization dynamics. The derivation is based on an eigenmode expansion of the associated refractory density equation, which generalizes previous spectral methods for Fokker-Planck equations to arbitrary renewal models. We find a simple relation between the eigenvalues, which determine the characteristic time scales of the firing rate dynamics, and the Laplace transform of the interspike interval density or the survival function of the renewal process. Analytical expressions for the Laplace transforms are readily available for many renewal models including the leaky integrate-and-fire model. Retaining only the first eigenmode yields already an adequate low-dimensional approximation of the firing-rate dynamics that captures spike synchronization effects and fast transient dynamics at stimulus onset. We explicitly demonstrate the validity of our model for a large homogeneous population of Poisson neurons with absolute refractoriness, and other renewal models that admit an explicit analytical calculation of the eigenvalues. The here presented eigenmode expansion provides a systematic framework for novel firing-rate models in computational neuroscience based on spiking neuron dynamics with refractoriness.Comment: 24 pages, 7 figure

Scaling behaviour in probabilistic neuronal cellular automata

We study a neural network model of interacting stochastic discrete two--state cellular automata on a regular lattice. The system is externally tuned to a critical point which varies with the degree of stochasticity (or the effective temperature). There are avalanches of neuronal activity, namely spatially and temporally contiguous sites of activity; a detailed numerical study of these activity avalanches is presented, and single, joint and marginal probability distributions are computed. At the critical point, we find that the scaling exponents for the variables are in good agreement with a mean--field theory.Comment: 7 pages, 4 figures Accepted for publication in Physical Review

Towards a theory of cortical columns: From spiking neurons to interacting neural populations of finite size

Neural population equations such as neural mass or field models are widely used to study brain activity on a large scale. However, the relation of these models to the properties of single neurons is unclear. Here we derive an equation for several interacting populations at the mesoscopic scale starting from a microscopic model of randomly connected generalized integrate-and-fire neuron models. Each population consists of 50 -- 2000 neurons of the same type but different populations account for different neuron types. The stochastic population equations that we find reveal how spike-history effects in single-neuron dynamics such as refractoriness and adaptation interact with finite-size fluctuations on the population level. Efficient integration of the stochastic mesoscopic equations reproduces the statistical behavior of the population activities obtained from microscopic simulations of a full spiking neural network model. The theory describes nonlinear emergent dynamics like finite-size-induced stochastic transitions in multistable networks and synchronization in balanced networks of excitatory and inhibitory neurons. The mesoscopic equations are employed to rapidly simulate a model of a local cortical microcircuit consisting of eight neuron types. Our theory establishes a general framework for modeling finite-size neural population dynamics based on single cell and synapse parameters and offers an efficient approach to analyzing cortical circuits and computations

Multiple forms of working memory emerge from synapse-astrocyte interactions in a neuron-glia network model

Persistent activity in populations of neurons, time-varying activity across a neural population, or activity-silent mechanisms carried out by hidden internal states of the neural population have been proposed as different mechanisms of working memory (WM). Whether these mechanisms could be mutually exclusive or occur in the same neuronal circuit remains, however, elusive, and so do their biophysical underpinnings. While WM is traditionally regarded to depend purely on neuronal mechanisms, cortical networks also include astrocytes that can modulate neural activity. We propose and investigate a network model that includes both neurons and glia and show that glia-synapse interactions can lead to multiple stable states of synaptic transmission. Depending on parameters, these interactions can lead in turn to distinct patterns of network activity that can serve as substrates for WM