Control Design and Analysis of a Stochastic Event-Driven System

Soltani, Mohammad; Singh, Abhyudai

doi:10.1109/cdc.2018.8619759

Cited by 5 publications

(3 citation statements)

References 47 publications

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“…In summary, we have defined a SHS model for synaptic transmission with continuous dynamics (13), two family of rests that occur with rates (3) and (11), and when they occur, the state space resets via (8) and (12), respectively. It is important to point out that this model with just the timer-dependent reset has been referred to in literature as time-triggered SHS [45][46][47][48]. Allowing for a second family of resets that occurs at a rate linearly dependent on x is a novelty of this work.…”

Section: Stochastic Hybrid System Modelmentioning

confidence: 99%

Characterizing neuronal synaptic transmission using stochastic hybrid systems

Vahdat

Singh

2019

Preprint

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Action potential-triggered release of neurotransmitters at chemical synapses forms the key basis of communication between two neurons. To quantify the stochastic dynamics of the number of neurotransmitters released, we investigate a model where neurotransmitter-filled vesicles attach to a finite number of docking sites in the axon terminal, and are subsequently released when the action potential arrives. We formulate the model as a Stochastic Hybrid System (SHS) that combines three key noise mechanisms: random arrival of action potentials, stochastic refilling of docking sites, and probabilistic release of docked vesicles. This SHS representation is used to derive exact analytical formulas for the mean and noise (as quantified by Fano factor) in the number of vesicles released per action potential. Interestingly, results show that in relevant parameter regimes, noise in the number of vesicles released is sub-Poissonian at low frequencies, super-Poissonian at intermediate frequencies, and approaches a Poisson limit at high frequencies.In contrast, noise in the number of neurotransmitters in the synaptic cleft is always super-Poissonian, but is lowest at intermediate frequencies.We further investigate changes in these noise properties for non-Poissonian arrival of action potentials, and when the probability of release is frequency dependent. In summary, these results provide the first glimpse into synaptic parameters not only determining the mean synaptic strength, but also shaping its stochastic dynamics that is critical for information transfer between neurons.rich literature regarding synaptic transmission as a deterministic process [1][2][3][4][5], increasing evidence points to diverse noise mechanisms at play during this process [6][7][8][9][10]. Counterintuitively, noise has been shown to sometimes enhance signaling and information transfer between neurons [11][12][13][14][15][16].Stochastic Hybrid Systems (SHS) constitute an important class of mathematical models that integrate discrete stochastic events with continuous dynamics. Given their generality and scope, SHS have been successfully used for modeling stochastic phenomena in a variety of biological processes [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33], including neuronal dynamics [34,35]. Building up on our previous work [36], we consider the SHS framework for modeling the neurotransmitter dynamics. More specifically, the model consists of M ∈ {1, 2, . . .} docking site at the axon terminal (Fig. 1). Neurotransmitter-filled vesicles attach to these docking sites with a given probabilistic rate that is proportional to M − n, where n is the number of already docked vesicles. Action Potentials (APs) arrive at a given frequency, such that, the time between two successive APs follows an arbitrary positively-valued probability density function g. Each docked vesicle has a certain probability of release, and based on this probability, APs cause a fraction of the docked vesicles to release their neurotransmitter content into the synapt...

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Section: Stochastic Hybrid System Modelmentioning

confidence: 99%

Characterizing neuronal synaptic transmission using stochastic hybrid systems

Vahdat

Singh

2019

Preprint

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“…Whenever the event occurs the state undergoes a random jump based on a given random map, and the timer is reset to zero. These class of SHS have been shown to be quite useful in modeling and analysis of networked control systems [20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Time-triggered stochastic hybrid systems with two timer-dependent resets

Singh¹,

Vahdat²,

Xu³

2019

Preprint

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We analyze a class of time-triggered stochastic hybrid systems where the state-space evolves as per a linear time-invariant dynamical system. This continuous time evolution is interspersed with two kinds of stochastic resets. The first reset occurs based on an internal timer that measures the time elapsed since it last occurred. Whenever the first reset occurs the states-space undergoes a random jump and the timer is reset to zero. The second reset occurs based on an arbitrary timer-depended rate, and whenever this reset fires, the state-space is changed based on a given random map. For this class of systems, we provide exact conditions that lead to finite statistic moments, and the corresponding exact analytical expressions for the first two moments. This framework is applied to study random fluctuations in the concentration of a protein in a growing cell. In the context of this example, the timer denotes the time elapsed since the cell was born, and the cell division event (first reset) is triggered based on a timer-dependent rate. The second reset corresponds to synthesis of the protein in stochastic bursts, and finally, during cell growth protein concentration continuously decrease due to dilution. Our analysis provides closed-form formulas for the noise in the protein concentration and leads to a striking result - for a constant (timer-independent) protein synthesis rate the noise in the protein concentration is invariant of the noise in the cell-cycle time. Finally, we provide a rigorous framework for investigating protein noise levels for different forms of time-dependent synthesis rates, as is the case for cell-cycle regulated genes inside the cell.

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“…Such systems have been referred to in literature as time-triggered stochastic hybrid systems (TTSHS) [1][2][3][4], and are an important sub-class of piecewisedeterministic Markov processes (PDMP) [5][6][7][8] with applications in different disciplines. For example, TTSHS have been shown to arise ubiquitously in networked control systems, where a dynamical system is controlled over a noisy communication network, and signals are received at discrete random times [9][10][11][12][13][14][15][16][17]. Other TTSHS applications include modeling disturbances in nanosensors [18], capturing stochastic effects in cellular biochemical processes [19][20][21][22], and neuroscience [23].…”

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confidence: 99%

Moment Analysis of Linear Time-Varying Dynamical Systems with Renewal Transitions

Soltani¹,

Singh²

2019

SIAM J. Control Optim.

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Stochastic dynamics of several systems can be modeled via piecewise deterministic time evolution of the state, interspersed by random discrete events. Within this general class of systems, we consider time-triggered stochastic hybrid systems (TTSHS), where the state evolves continuously according to a linear time-varying dynamical system. Discrete events occur based on an underlying renewal process (timer), and the intervals between successive events follow an arbitrary continuous probability density function. Moreover, whenever the event occurs, the state is reset based on a linear affine transformation that allows for the inclusion of state-dependent and independent noise terms. Our key contribution is derivation of necessary and sufficient conditions for the stability of statistical moments, along with exact analytical expressions for the steady-state moments. These results are illustrated on an example from cell biology, where deterministic synthesis and decay of a gene product (RNA or protein) is coupled to random timing of cell-division events. As experimentally observed, cell-division events occur based on an internal timer that measures the time elapsed since the start of cell cycle (i.e., last event). Upon division, the gene product level is halved, together with a state-dependent noise term that arises due to randomness in the partitioning of molecules between two daughter cells. We show that the TTSHS framework is conveniently suited to capture the time evolution of gene product levels, and derive unique formulas connecting its mean and variance to underlying model parameters and noise mechanisms. Systematic analysis of the formulas reveal counterintuitive insights, such as, if the partitioning noise is large then making the timing of cell division more random reduces noise in gene product levels. In summary, theory developed here provides novel tools for characterizing moments in an important class of stochastic dynamical systems that arises naturally in diverse application areas.

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Control Design and Analysis of a Stochastic Event-Driven System

Cited by 5 publications

References 47 publications

Characterizing neuronal synaptic transmission using stochastic hybrid systems

Characterizing neuronal synaptic transmission using stochastic hybrid systems

Time-triggered stochastic hybrid systems with two timer-dependent resets

Moment Analysis of Linear Time-Varying Dynamical Systems with Renewal Transitions

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