Measuring synchronous bursting and spiking under varying second order network connectivity statistics

Synchronous bursting plays an integral role in a variety of applications, from generating respiratory rhythms and inducing hormonal releases to conveying information about a stimulus. Since the network structure influences the dynamical behavior of a network, we aim to identify network structures that promote synchronous bursting.\ud \ud By generating the network topology using specific probability distributions, Zhao et al. [1] demonstrated that the prevalence of certain two-connection network motifs significantly affects synchronous spiking. We build on these results by constructing a new graph theoretical method for quantifying how the network topology promotes synchronous spiking and bursting. We first verify the validity of the new measure by establishing consistency with the results of Zhao et al. By utilizing this new measure, our gains are two-fold. This new measure not only generalizes prior results on synchronous spiking, as the framework can accommodate network topologies generated by any probability distribution, but also provides a deterministic method for quantifying the likelihood of synchronous bursting under various network topologies.\ud \ud We construct the graph theoretical measure by restricting the numbers of incoming and outgoing connections of a neuron to take one of finitely many possible values. Fixing certain constraints, such as the number of neurons in our network and the expected number of connections, we construct a probability distribution for our network. Through averaging, we construct difference equations from the probability distribution to track the expected number of active neurons and inactive neurons in a given time step. In the special case of spiking, the difference equations force a neuron to fire if one of its neighbors fires. By considering the maximum number of active neurons within a single time step, we formulate a graph theoretical measure for synchronous spiking. Using an analogous system of difference equations, we can construct a graph theoretical measure for synchronous bursting as well. Consequently, by calculating the second order network statistics from the probability distribution, and identifying all candidate probability distributions that satisfy the constrains, we can analyze how second order network statistics from any probability distribution promote synchronous spiking and bursting.\ud \ud In our simulations, we find that increasing the covariance of the in-degree and out-degree monotonically increases the predicted occurrence of synchronous spiking under the graph theoretic measure. We also note that for a wide range of parameters, the number of neurons in the network and the constraints on the governing probability distribution of our network have minimal qualitative impact on the relationship between second order network statistics and predicted likelihood of synchronous spiking. Furthermore, preliminary results regarding the effect of second order network statistics on predicted synchronous bursting suggest a more intricate relationship than in the case of synchronous spiking. Based on the consistency of the measure in the case of synchronous spiking with the existing literature and due to its ability to incorporate relatively abstract properties of the network, we conjecture that the measure will provide new insight regarding the impact of the network topology on continuous time-scale models of synchronous bursting and other complex behaviors

Information transfer in signaling pathways : a study using coupled simulated and experimental data

Background: The topology of signaling cascades has been studied in quite some detail. However, how information is processed exactly is still relatively unknown. Since quite diverse information has to be transported by one and the same signaling cascade (e.g. in case of different agonists), it is clear that the underlying mechanism is more complex than a simple binary switch which relies on the mere presence or absence of a particular species. Therefore, finding means to analyze the information transferred will help in deciphering how information is processed exactly in the cell. Using the information-theoretic measure transfer entropy, we studied the properties of information transfer in an example case, namely calcium signaling under different cellular conditions. Transfer entropy is an asymmetric and dynamic measure of the dependence of two (nonlinear) stochastic processes. We used calcium signaling since it is a well-studied example of complex cellular signaling. It has been suggested that specific information is encoded in the amplitude, frequency and waveform of the oscillatory Ca2+-signal. Results: We set up a computational framework to study information transfer, e.g. for calcium signaling at different levels of activation and different particle numbers in the system. We stochastically coupled simulated and experimentally measured calcium signals to simulated target proteins and used kernel density methods to estimate the transfer entropy from these bivariate time series. We found that, most of the time, the transfer entropy increases with increasing particle numbers. In systems with only few particles, faithful information transfer is hampered by random fluctuations. The transfer entropy also seems to be slightly correlated to the complexity (spiking, bursting or irregular oscillations) of the signal. Finally, we discuss a number of peculiarities of our approach in detail. Conclusion: This study presents the first application of transfer entropy to biochemical signaling pathways. We could quantify the information transferred from simulated/experimentally measured calcium signals to a target enzyme under different cellular conditions. Our approach, comprising stochastic coupling and using the information-theoretic measure transfer entropy, could also be a valuable tool for the analysis of other signaling pathways

Information transfer in signaling pathways : a study using coupled simulated and experimental data

Background: The topology of signaling cascades has been studied in quite some detail. However, how information is processed exactly is still relatively unknown. Since quite diverse information has to be transported by one and the same signaling cascade (e.g. in case of different agonists), it is clear that the underlying mechanism is more complex than a simple binary switch which relies on the mere presence or absence of a particular species. Therefore, finding means to analyze the information transferred will help in deciphering how information is processed exactly in the cell. Using the information-theoretic measure transfer entropy, we studied the properties of information transfer in an example case, namely calcium signaling under different cellular conditions. Transfer entropy is an asymmetric and dynamic measure of the dependence of two (nonlinear) stochastic processes. We used calcium signaling since it is a well-studied example of complex cellular signaling. It has been suggested that specific information is encoded in the amplitude, frequency and waveform of the oscillatory Ca2+-signal. Results: We set up a computational framework to study information transfer, e.g. for calcium signaling at different levels of activation and different particle numbers in the system. We stochastically coupled simulated and experimentally measured calcium signals to simulated target proteins and used kernel density methods to estimate the transfer entropy from these bivariate time series. We found that, most of the time, the transfer entropy increases with increasing particle numbers. In systems with only few particles, faithful information transfer is hampered by random fluctuations. The transfer entropy also seems to be slightly correlated to the complexity (spiking, bursting or irregular oscillations) of the signal. Finally, we discuss a number of peculiarities of our approach in detail. Conclusion: This study presents the first application of transfer entropy to biochemical signaling pathways. We could quantify the information transferred from simulated/experimentally measured calcium signals to a target enzyme under different cellular conditions. Our approach, comprising stochastic coupling and using the information-theoretic measure transfer entropy, could also be a valuable tool for the analysis of other signaling pathways

Extreme phase sensitivity in systems with fractal isochrons

Sensitivity to initial conditions is usually associated with chaotic dynamics and strange attractors. However, even systems with (quasi)periodic dynamics can exhibit it. In this context we report on the fractal properties of the isochrons of some continuous-time asymptotically periodic systems. We define a global measure of phase sensitivity that we call the phase sensitivity coefficient and show that it is an invariant of the system related to the capacity dimension of the isochrons. Similar results are also obtained with discrete-time systems. As an illustration of the framework, we compute the phase sensitivity coefficient for popular models of bursting neurons, suggesting that some elliptic bursting neurons are characterized by isochrons of high fractal dimensions and exhibit a very sensitive (unreliable) phase response.Comment: 32 page