Learning Brain Dynamics With Coupled Low-Dimensional Nonlinear Oscillators and Deep Recurrent Networks.

Many natural systems, especially biological ones, exhibit complex multivariate nonlinear dynamical behaviors that can be hard to capture by linear autoregressive models. On the other hand, generic nonlinear models such as deep recurrent neural networks often require large amounts of training data, not always available in domains such as brain imaging; also, they often lack interpretability. Domain knowledge about the types of dynamics typically observed in such systems, such as a certain type of dynamical systems models, could complement purely data-driven techniques by providing a good prior. In this work, we consider a class of ordinary differential equation (ODE) models known as van der Pol (VDP) oscil lators and evaluate their ability to capture a low-dimensional representation of neural activity measured by different brain imaging modalities, such as calcium imaging (CaI) and fMRI, in different living organisms: larval zebrafish, rat, and human. We develop a novel and efficient approach to the nontrivial problem of parameters estimation for a network of coupled dynamical systems from multivariate data and demonstrate that the resulting VDP models are both accurate and interpretable, as VDP's coupling matrix reveals anatomically meaningful excitatory and inhibitory interactions across different brain subsystems. VDP outperforms linear autoregressive models (VAR) in terms of both the data fit accuracy and the quality of insight provided by the coupling matrices and often tends to generalize better to unseen data when predicting future brain activity, being comparable to and sometimes better than the recurrent neural networks (LSTMs). Finally, we demonstrate that our (generative) VDP model can also serve as a data-augmentation tool leading to marked improvements in predictive accuracy of recurrent neural networks. Thus, our work contributes to both basic and applied dimensions of neuroimaging: gaining scientific insights and improving brain-based predictive models, an area of potentially high practical importance in clinical diagnosis and neurotechnology.

Quantification of fluctuations from fluorescence correlation spectroscopy experiments in reaction-diffusion systems.

Fluorescence correlation spectroscopy (FCS) is commonly used to estimate diffusion and reaction rates. In FCS the fluorescence coming from a small volume is recorded and the autocorrelation function (ACF) of the fluorescence fluctuations is computed. Scaling out the fluctuations due to the emission process, this ACF can be related to the ACF of the fluctuations in the number of observed fluorescent molecules. In this paper the ACF of the molecule number fluctuations is studied theoretically, with no approximations, for a reaction-diffusion system in which the fluorescence changes with binding and unbinding. Theoretical ACFs are usually derived assuming that fluctuations in the number of molecules of one species are instantaneously uncorrelated to those of the others and obey Poisson statistics. Under these assumptions, the ACF derived in this paper is characterized only by the diffusive timescale of the fluorescent species and its total weight is the inverse of the mean number of observed fluorescent molecules. The theory is then scrutinized in view of previous experimental results which, for a similar system, gave a different total weight and correct estimates of other diffusive timescales. The total weight mismatch is corrected by assuming that the variance of the number of fluorescent molecules depends on the variance of the particle numbers of the other species, as in the variance decomposition formula. Including the finite acquisition time in its computation, it is shown that the ACF depends on various timescales of the system and that its total weight coincides with the one obtained with the variance decomposition formula. This calculation implies that diffusion coefficients of nonobservable species can be estimated with FCS experiments performed in reaction-diffusion systems. Ways to proceed in future experiments are also discussed.

The effect of reactions on the formation and readout of the gradient of Bicoid.

During early development, the establishment of gradients of transcriptional factors determines the patterning of cell fates. The case of Bicoid (Bcd) in Drosophila melanogaster embryos is well documented and studied. There are still controversies as to whether SDD models in which Bcd is Synthesized at one end, then Diffuses and is Degraded can explain the gradient formation within the timescale observed experimentally. The Bcd gradient is observed in embryos that express a Bicoid-eGFP fusion protein (Bcd-GFP) which cannot differentiate if Bcd is freely diffusing or bound to immobile sites. In this work we analyze an SDID model that includes the Interaction of Bcd with binding sites. We simulate numerically the resulting full reaction-diffusion system in a cylindrical domain using previously determined biophysical parameters and a simplified version of the Bcd source. In this way we obtain solutions that depend on the spatial location approximately as observed experimentally and that reach such dependence at a time that is also compatible with the experimental observations. Analyzing the differences between the free and bound Bcd distributions we observe that the latter spans over a longer lengthscale. We conclude that deriving the lengthscale from the distribution of Bcd-GFP can lead to an overestimation of the gradient lengthscale and of the Hill coefficient that relates the concentrations of Bcd and of the protein, Hunchback, whose production is regulated by Bcd.

Sexual pheromone modulates the frequency of cytosolic Ca2+ bursts in Saccharomyces cerevisiae.

Transient and highly regulated elevations of cytosolic Ca2+ control a variety of cellular processes. Bulk measurements using radioactive Ca2+ and the luminescent sensor aequorin have shown that in response to pheromone, budding yeast cells undergo a rise of cytosolic Ca2+ that is mediated by two import systems composed of the Mid1-Cch1-Ecm7 protein complex and the Fig1 protein. Although this response has been widely studied, there is no treatment of Ca2+ dynamics at the single-cell level. Here, using protein calcium indicators, we show that both vegetative and pheromone-treated yeast cells exhibit discrete and asynchronous Ca2+ bursts. Most bursts reach maximal amplitude in 1-10 s, range between 7 and 30 s, and decay in a way that fits a single-exponential model. In vegetative cells, bursts are scarce but preferentially occur when cells are transitioning G1 and S phases. On pheromone presence, Ca2+ burst occurrence increases dramatically, persisting during cell growth polarization. Pheromone concentration modulates burst frequency in a mechanism that depends on Mid1, Fig1, and a third, unidentified, import system. We also show that the calcineurin-responsive transcription factor Crz1 undergoes nuclear localization bursts during the pheromone response.

Hindered cytoplasmic diffusion of inositol trisphosphate restricts its cellular range of action.

The range of action of intracellular messengers is determined by their rates of diffusion and degradation. Previous measurements in oocyte cytoplasmic extracts indicated that the Ca2+-liberating second messenger inositol trisphosphate (IP3) diffuses with a coefficient (~280 µm2 s-1) similar to that in water, corresponding to a range of action of ~25 µm. Consequently, IP3 is generally considered a "global" cellular messenger. We reexamined this issue by measuring local IP3-evoked Ca2+ puffs to monitor IP3 diffusing from spot photorelease in neuroblastoma cells. Fitting these data by numerical simulations yielded a diffusion coefficient (≤10 µm2 s-1) about 30-fold slower than that previously reported. We propose that diffusion of IP3 in mammalian cells is hindered by binding to immobile, functionally inactive receptors that were diluted in oocyte extracts. The predicted range of action of IP3 (<5 µm) is thus smaller than the size of typical mammalian cells, indicating that IP3 should better be considered as a local rather than a global cellular messenger.

Luminal Ca(2+) dynamics during IP3R mediated signals.

The role of cytosolic Ca(2+) on the kinetics of Inositol 1,4,5-triphosphate receptors (IP3Rs) and on the dynamics of IP3R-mediated Ca(2+) signals has been studied at large both experimentally and by modeling. The role of luminal Ca(2+) has not been investigated with that much detail although it has been found that it is relevant for signal termination in the case of Ca(2+) release through ryanodine receptors. In this work we present the results of observing the dynamics of luminal and cytosolic Ca(2+) simultaneously in Xenopus laevis oocytes. Combining observations and modeling we conclude that there is a rapid mechanism that guarantees the availability of free Ca(2+) in the lumen even when a relatively large Ca(2+) release is evoked. Comparing the dynamics of cytosolic and luminal Ca(2+) during a release, we estimate that they are consistent with a 80% of luminal Ca(2+) being buffered. The rapid availability of free luminal Ca(2+) correlates with the observation that the lumen occupies a considerable volume in several regions across the images.

Buffer regulation of calcium puff sequences.

Puffs are localized Ca(2 +) signals that arise in oocytes in response to inositol 1,4,5-trisphosphate (IP3). They are the result of the liberation of Ca(2 +) from the endoplasmic reticulum through the coordinated opening of IP3 receptor/channels clustered at a functional release site. The presence of buffers that trap Ca(2 +) provides a mechanism that enriches the spatio-temporal dynamics of cytosolic calcium. The expression of different types of buffers along the cell's life provides a tool with which Ca(2 +) signals and their responses can be modulated. In this paper we extend the stochastic model of a cluster of IP3R-Ca(2 +) channels introduced previously to elucidate the effect of buffers on sequences of puffs at the same release site. We obtain analytically the probability laws of the interpuff time and of the number of channels that participate of the puffs. Furthermore, we show that under typical experimental conditions the effect of buffers can be accounted for in terms of a simple inhibiting function. Hence, by exploring different inhibiting functions we are able to study the effect of a variety of buffers on the puff size and interpuff time distributions. We find the somewhat counter-intuitive result that the addition of a fast Ca(2 +) buffer can increase the average number of channels that participate of a puff.

From free to effective diffusion coefficients in fluorescence correlation spectroscopy experiments.

Diffusion is one of the main transport processes that occur inside cells determining the spatial and time distribution of relevant action molecules. In most cases these molecules not only diffuse but also interact with others as they get transported. When these interactions occur faster than diffusion the resulting transport can be characterized by "effective diffusion coefficients" that depend on both the reaction rates and the "free" diffusion coefficients. Fluorescence correlation spectroscopy (FCS) gives information on effective rather than free diffusion coefficients under this condition. In the present paper we investigate what coefficients can be drawn from FCS experiments for a wide range of values of the ratio of reaction to diffusion time scales, using different fitting functions. We find that the effective coefficients can be inferred with relatively small errors even when the condition of fast reactions does not exactly hold. Since the diffusion time scale depends on the size of the observation volume and the reaction time scale depends on concentrations, we also discuss how by changing either one or the other property one can switch between the two limits and extract more information on the system under study.

Mean field strategies induce unrealistic non-linearities in calcium puffs.

Mean field models are often useful approximations to biological systems, but sometimes, they can yield misleading results. In this work, we compare mean field approaches with stochastic models of intracellular calcium release. In particular, we concentrate on calcium signals generated by the concerted opening of several clustered channels (calcium puffs). To this end we simulate calcium puffs numerically and then try to reproduce features of the resulting calcium distribution using mean field models were all the channels open and close simultaneously. We show that an unrealistic non-linear relationship between the current and the number of open channels is needed to reproduce the simulated puffs. Furthermore, a single channel current which is five times smaller than the one of the stochastic simulations is also needed. Our study sheds light on the importance of the stochastic kinetics of the calcium release channel activity to estimate the release fluxes.

Custom-made modification of a commercial confocal microscope to photolyze caged compounds using the conventional illumination module and its application to the observation of Inositol 1,4,5-trisphosphate-mediated calcium signals.

The flash photolysis of "caged" compounds is a powerful experimental technique for producing rapid changes in concentrations of bioactive signaling molecules. These caged compounds are inactive and become active when illuminated with ultraviolet light. This paper describes an inexpensive adaptation of an Olympus confocal microscope that uses as source of ultraviolet light the mercury lamp that comes with the microscope for conventional fluorescence microscopy. The ultraviolet illumination from the lamp (350 - 400 nm) enters through an optical fiber that is coupled to a nonconventional port of the microscope. The modification allows to perform the photolysis of caged compounds over wide areas (∼ 200 µm) and obtain confocal fluorescence images simultaneously. By controlling the ultraviolet illumination exposure time and intensity it is possible to regulate the amount of photolyzed compounds. In the paper we characterize the properties of the system and show its capabilities with experiments done in aqueous solution and in Xenopus Laevis oocytes. The latter demonstrate its applicability for the study of Inositol 1,4,5-trisphosphate-mediated intracellular calcium signals.

Intra-cluster percolation of calcium signals.

Calcium signals are involved in a large variety of physiological processes. Their versatility relies on the diversity of spatiotemporal behaviors that the calcium concentration can display. Calcium entry through inositol 1,4,5-trisphosphate (IP3) receptors (IP3R's) is a key component that participates in both local signals such as "puffs" and in global waves. IP3R's areusually organized in clusters on the membrane of the endoplasmic reticulum and their spatial distribution has important effects on the resulting signal. Recent high resolution observations of Ca2+ puffs offer a window to study intra-cluster organization. The experiments give the distribution of the number of IP3R's that open during each puff without much processing. Here we present a simple model with which we interpret the experimental distribution in terms of two stochastic processes: IP3 binding and unbinding and Ca2+-mediated inter-channel coupling. Depending on the parameters of the system, the distribution may be dominated by one or the other process. The transition between both extreme case sis similar to a percolation process. We show how, from an analysis of the experimental distribution, information can be obtained on the relative weight of the two processes. The largest distance over which Ca2+-mediated coupling acts and the density of IP3-bound IP3R's of the cluster can also be estimated. The approach allows us to infer properties of the interactions among the channels of the cluster from statistical information on their emergent collective behavior.

Quantifying calcium fluxes underlying calcium puffs in Xenopus laevis oocytes.

We determine the calcium fluxes through inositol 1,4,5-trisphosphate receptor/channels underlying calcium puffs of Xenopus laevis oocytes using a simplified version of the algorithm of Ventura et al. An analysis of 130 puffs obtained with Fluo-4 indicates that Ca2+ release comes from a region of width approximately 450 nm, that the release duration is peaked around 18 s and that the underlying Ca2+ currents range between 0.12 and 0.95 pA. All these parameters are independent of IP(3) concentration. We explore what distributions of channels that open during a puff, N(p), and what relations between current and number of open channels, I(N(p)), are compatible with our findings and with the distribution of puff-to-trigger amplitude ratio reported in Rose et al. To this end, we use simple "mean field" models in which all channels open and close simultaneously. We find that the variability among clusters plays an important role in shaping the observed puff amplitude distribution and that a model for which I(N(p)) approximately N(p) for small N(p) and I(N(p)) approximately N(p)(1/alpha) (alpha > 1) for large N(p), provides the best agreement. Simulations of more detailed models in which channels open and close stochastically show that this nonlinear behavior can be attributed to the limited time resolution of the observations and to the averaging procedure that is implicit in the mean-field models. These conclusions are also compatible with observations of approximately 400 puffs obtained using the dye Oregon green.

Optical techniques provide information on various effective diffusion coefficients in the presence of traps.

In many cell-signaling pathways information is transmitted via the diffusion of messenger molecules. In most cases, messengers react with other substances and diffuse at the same time. Effective diffusion coefficients may be introduced to characterize the net transport rate that results from the combined effect of these two processes. It was shown in [B. Pando, Proc. Natl. Acad. Sci. U.S.A. 103, 5338 (2006)] that even in the simplest scenario in which one bimolecular reaction is involved, two different effective coefficients are relevant. One gives the rate at which small perturbations spread out with time while the other relates the mean square displacement of a single particle to the time elapsed. They coincide in the absence of reactions but may be very different in other cases. Optical techniques provide a relatively noninvasive means by which transport rates can be estimated. In the above mentioned paper it was discussed why, under certain conditions, fluorescence recovery after photobleaching (FRAP), a technique commonly used to estimate diffusion rates in cells, provides information on one of the two effective coefficients. In the present paper we show that, under the same conditions, another commonly used optical technique, fluorescence correlation spectroscopy (FCS), gives information on the other one. This opens up the possibility of combining experiments to obtain information that goes beyond effective transport rates. In the present paper we discuss different ways to do so.

Simple data-driven models of intracellular calcium dynamics with predictive power.

Biology is complex. However, it is not clear how much of this complexity must necessarily translate into complicated mathematical models of biological processes. Simple models can be appealing to physicists but are usually deceiving for biologists. Complicated models, on the other hand, depend on too many parameters whose values are frequently unknown. Therefore, complicated models, although in principle more realistic, can lead to erroneous results if they are sensitive to these unknown parameter values. Intracellular calcium signals provide an example of utmost biological importance in which the issue of "simple vs complex" can be explored. In this paper we show that simple models describing the dynamics of intracellular calcium can be directly inferred from experimental data, without no a priori information on unknown parameters. A similar approach can be followed to study other reaction-diffusion systems. In spite of their simplicity, these models can provide quantitative information on some of the processes that shape calcium signals, such as the calcium current that underlies an experimental observation. This shows that simple models of biological systems are not limited to qualitative descriptions.

Messages diffuse faster than messengers.

In many cell-signaling pathways, information is transmitted by the diffusion of messenger molecules. Diffusion coefficients characterize the messenger's spatial range and the characteristic times of signal propagation. Inside cells, particles usually diffuse in the presence of immobile binding sites (or traps). It is well known that binding to traps results in an effective diffusion coefficient that is smaller than the free coefficient in media free of traps. To measure effective diffusion coefficients in cells, "tagged" particles are often used. Radioactive calcium was used in a giant squid axon and in cytosolic extracts of Xenopus laevis oocytes. Fluorescence recovery after photobleaching yields diffusion coefficients from observations of the distribution of fluorescently labeled proteins. In the absence of traps, free diffusion coefficients give both the rate at which single-particle mean square displacements increase and the rate at which information in the form of inhomogeneities in particle concentration spread out with time. We show here that, in the presence of traps, information diffuses faster than single particles. Thus, messages diffuse faster than messengers. Tagged-particle experiments give the single-particle diffusion coefficients and, thus, can underestimate the rate of diffusive signal propagation.

Analysis of puff dynamics in oocytes: interdependence of puff amplitude and interpuff interval.

Puffs are localized Ca(2+) signals that arise in oocytes in response to inositol 1,4,5-trisphosphate (IP(3)). They are analogous to the sparks of myocytes and are believed to be the result of the liberation of Ca(2+) from the endoplasmic reticulum through the coordinated opening of IP(3) receptor/channels clustered at a functional release site. In this article, we analyze sequences of puffs that occur at the same site to help elucidate the mechanisms underlying puff dynamics. In particular, we show a dependence of the interpuff time on the amplitude of the preceding puff, and of the amplitude of the following puff on the preceding interval. These relationships can be accounted for by an inhibitory role of the Ca(2+) that is liberated during puffs. We construct a stochastic model for a cluster of IP(3) receptor/channels that quantitatively replicates the observed behavior, and we determine that the characteristic time for a channel to escape from the inhibitory state is of the order of seconds.

A model-independent algorithm to derive Ca2+ fluxes underlying local cytosolic Ca2+ transients.

Local intracellular Ca(2+) signals result from Ca(2+) flux into the cytosol through individual channels or clusters of channels. To gain a mechanistic understanding of these events we need to know the magnitude and spatial distribution of the underlying Ca(2+) flux. However, this is difficult to infer from fluorescence Ca(2+) images because the distribution of Ca(2+)-bound dye is affected by poorly characterized processes including diffusion of Ca(2+) ions, their binding to mobile and immobile buffers, and sequestration by Ca(2+) pumps. Several methods have previously been proposed to derive Ca(2+) flux from fluorescence images, but all require explicit knowledge or assumptions regarding these processes. We now present a novel algorithm that requires few assumptions and is largely model-independent. By testing the algorithm with both numerically generated image data and experimental images of sparklets resulting from Ca(2+) flux through individual voltage-gated channels, we show that it satisfactorily reconstructs the magnitude and time course of the underlying Ca(2+) currents.

Role of complexing agents in the appearance of Turing patterns.

In this paper we study a four-species reaction-diffusion system where Turing patterns are stabilized by the presence of fast reversible reactions between the morphogens and two different mobile complexing agents (CAs) that are not necessarily in excess. We provide a quantitative explanation of how the interaction with the CA changes the size of the Turing space making it possible to observe patterns even in a region where the free diffusion coefficients of the relevant species are equal, as is usually the case in real systems. Our analytical treatment gives a series of mathematical relations that can be helpful for those designing experiments where Turing patterns are expected to appear. We also show how the mobility of CAs affect the characteristic size of the pattern. Finally, we provide an example of biological interest in order to illustrate the main procedures and results.

A model of IP3 receptor with a luminal calcium binding site: stochastic simulations and analysis.

We have constructed a stochastic model of the inositol 1,4,5-trisphosphate receptor-Ca2+ channel that is based on quantitative measurements of the channel's properties. It displays the observed dependence of the open probability of the channel with cytosolic [Ca2+] and [IP3] and gives values for the dwell times that agree with the observations. The model includes an explicit dependence of channel gating with luminal calcium. This not only explains several observations reported in the literature, but also provides a possible explanation of why the open probabilities and shapes of the bell-shaped curves reported in [Nature 351 (1991) 751] and in [Proc. Natl. Acad. Sci. U.S.A. 269 (1998) 7238] are so different.

Saltatory and continuous calcium waves and the rapid buffering approximation.

Calcium waves propagate inside cells due to a regenerative mechanism known as calcium-induced calcium release. Buffer-mediated calcium diffusion in the cytosol plays a crucial role in the process. However, most models of calcium waves either treat buffers phenomenologically or assume that they are in equilibrium with calcium (the rapid buffering approximation). In this article we address the issue of whether this approximation provides a good description of wave propagation. We first compare the timescales present in the problem, and determine the situations in which the equilibrium hypothesis fails. We then present a series of numerical studies based on the simple fire-diffuse-fire model of wave propagation. We find that the differences between the full and reduced descriptions may lead to errors that are above experimental resolution even for relatively fast buffers in the case of saltatory waves. Conversely, in the case of continuous waves, the approximation may give accurate results even for relatively slow buffers.