Natalia Z. Bielczyk

Dynamical Models of Dyadic Interactions with Delay

We investigate a general class of linear models of dyadic interactions with a constant discrete time delay. We prove that the changes in stability of the stationary states occur for various intervals of the parameters that determine the strength and nature of emotional interactions between the partners. The dynamics of interactions depend on both reactivity of partners to their own emotional states as well as to the partners states. The results suggest that reactivity to the partners states has greater impact on the dynamics of the relationship than the reactivity to ones own states. Moreover, the results underscore the importance of deliberation in maintaining the stability of the relationship. Moreover, we have found that multiple stability switches are only possible when one of the partners reacts with delay to their own emotional states. We also propose a generalization to triadic interactions.

Delay can stabilize: Love affairs dynamics

Abstract We discuss two models of interpersonal interactions with delay. The first model is linear, and allows the presentation of a rigorous mathematical analysis of stability, while the second is nonlinear and a typical local stability analysis is thus performed. The linear model is a direct extension of the classic Strogatz model. On the other hand, as interpersonal relations are nonlinear dynamical processes, the nonlinear model should better reflect real interactions. Both models involve immediate reaction on partner’s state and a correction of the reaction after some time. The models we discuss belong to the class of two-variable systems with one delay for which appropriate delay stabilizes an unstable steady state. We formulate a theorem and prove that stabilization takes place in our case. We conclude that considerable (meaning large enough, but not too large) values of time delay involved in the model can stabilize love affairs dynamics.

Circuit to construct mapping: a mathematical tool for assisting the diagnosis and treatment in major depressive disorder.

Major depressive disorder (MDD) is a serious condition with a lifetime prevalence exceeding 16% worldwide. MDD is a heterogeneous disorder that involves multiple behavioral symptoms on the one hand and multiple neuronal circuits on the other hand. In this review, we integrate the literature on cognitive and physiological biomarkers of MDD with the insights derived from mathematical models of brain networks, especially models that can be used for fMRI datasets. We refer to the recent NIH research domain criteria initiative, in which a concept of “constructs” as functional units of mental disorders is introduced. Constructs are biomarkers present at multiple levels of brain functioning – cognition, genetics, brain anatomy, and neurophysiology. In this review, we propose a new approach which we called circuit to construct mapping (CCM), which aims to characterize causal relations between the underlying network dynamics (as the cause) and the constructs referring to the clinical symptoms of MDD (as the effect). CCM involves extracting diagnostic categories from behavioral data, linking circuits that are causal to these categories with use of clinical neuroimaging data, and modeling the dynamics of the emerging circuits with attractor dynamics in order to provide new, neuroimaging-related biomarkers for MDD. The CCM approach optimizes the clinical diagnosis and patient stratification. It also addresses the recent demand for linking circuits to behavior, and provides a new insight into clinical treatment by investigating the dynamics of neuronal circuits underneath cognitive dimensions of MDD. CCM can serve as a new regime toward personalized medicine, assisting the diagnosis and treatment of MDD.

Thresholding functional connectomes by means of mixture modeling

&NA; Functional connectivity has been shown to be a very promising tool for studying the large‐scale functional architecture of the human brain. In network research in fMRI, functional connectivity is considered as a set of pair‐wise interactions between the nodes of the network. These interactions are typically operationalized through the full or partial correlation between all pairs of regional time series. Estimating the structure of the latent underlying functional connectome from the set of pair‐wise partial correlations remains an open research problem though. Typically, this thresholding problem is approached by proportional thresholding, or by means of parametric or non‐parametric permutation testing across a cohort of subjects at each possible connection. As an alternative, we propose a data‐driven thresholding approach for network matrices on the basis of mixture modeling. This approach allows for creating subject‐specific sparse connectomes by modeling the full set of partial correlations as a mixture of low correlation values associated with weak or unreliable edges in the connectome and a sparse set of reliable connections. Consequently, we propose to use alternative thresholding strategy based on the model fit using pseudo‐False Discovery Rates derived on the basis of the empirical null estimated as part of the mixture distribution. We evaluate the method on synthetic benchmark fMRI datasets where the underlying network structure is known, and demonstrate that it gives improved performance with respect to the alternative methods for thresholding connectomes, given the canonical thresholding levels. We also demonstrate that mixture modeling gives highly reproducible results when applied to the functional connectomes of the visual system derived from the n‐back Working Memory task in the Human Connectome Project. The sparse connectomes obtained from mixture modeling are further discussed in the light of the previous knowledge of the functional architecture of the visual system in humans. We also demonstrate that with use of our method, we are able to extract similar information on the group level as can be achieved with permutation testing even though these two methods are not equivalent. We demonstrate that with both of these methods, we obtain functional decoupling between the two hemispheres in the higher order areas of the visual cortex during visual stimulation as compared to the resting state, which is in line with previous studies suggesting lateralization in the visual processing. However, as opposed to permutation testing, our approach does not require inference at the cohort level and can be used for creating sparse connectomes at the level of a single subject. HighlightsSparse functional connectomes are useful in analyzing and interpreting fMRI data.We propose thresholding by means of mixture modeling and control of FDR.We benchmark the approach on synthetic fMRI data against established methods.We apply the method to the resting state and working memory task datasets from HCP500.Results are reproducible on synthetic data and interpretable on experimental data.

Time-delay model of perceptual decision making in cortical networks

It is known that cortical networks operate on the edge of instability, in which oscillations can appear. However, the influence of this dynamic regime on performance in decision making, is not well understood. In this work, we propose a population model of decision making based on a winner-take-all mechanism. Using this model, we demonstrate that local slow inhibition within the competing neuronal populations can lead to Hopf bifurcation. At the edge of instability, the system exhibits ambiguity in the decision making, which can account for the perceptual switches observed in human experiments. We further validate this model with fMRI datasets from an experiment on semantic priming in perception of ambivalent (male versus female) faces. We demonstrate that the model can correctly predict the drop in the variance of the BOLD within the Superior Parietal Area and Inferior Parietal Area while watching ambiguous visual stimuli. Author summary Human cortex is a complex structure composed of thousands of tangled neural circuits. These circuits exhibit multiple modes of activity, depending on the local balance between excitatory and inhibitory activity. In particular, these circuits can exhibit oscillatory behavior, which is believed to be a manifestation of a so-called criticality: balancing on the edge between stable and unstable dynamics. Circuits in the cortex are responsible for higher cognitive functions such as, in example, perceptual decision making, i.e., evaluating properties of objects appearing in the visual field. However, it is not well known how aforementioned balancing on the edge of instability influences perceptual decision making. In this work, we build a model to simulate dynamics of a very simple decision-making network consisting of two subpopulations. We then demonstrate that criticality in the network can account for ambiguity in decision making, and cause perceptual switches observed in human experiments. We further validate our model with datasets coming from a functional Magnetic Resonance Imaging experiment on semantic priming in perception of ambivalent (male versus female) faces. We demonstrate that the model can correctly predict the drop in the variance of the BOLD within the parietal areas of the cortex while watching ambiguous visual stimuli.

Disentangling casual webs in the brain using functional magnetic resonance imaging: A review of current approaches

In the past two decades, functional Magnetic Resonance Imaging (fMRI) has been used to relate neuronal network activity to cognitive processing and behavior. Recently this approach has been augmented by algorithms that allow us to infer causal links between component populations of neuronal networks. Multiple inference procedures have been proposed to approach this research question but so far, each method has limitations when it comes to establishing whole-brain connectivity patterns. In this paper, we discuss eight ways to infer causality in fMRI research: Bayesian Nets, Dynamical Causal Modelling, Granger Causality, Likelihood Ratios, Linear Non-Gaussian Acyclic Models, Patel’s Tau, Structural Equation Modelling, and Transfer Entropy. We finish with formulating some recommendations for the future directions in this area.

Cortical control of aggression: GABA signalling in the anterior cingulate cortex

Reduced top-down control by cortical areas is assumed to underlie pathological forms of aggression. While the precise underlying molecular mechanisms are still elusive, it seems that balancing the excitatory and inhibitory tones of cortical brain areas has a role in aggression control. The molecular mechanisms underpinning aggression control were examined in the BALB/cJ mouse model. First, these mice were extensively phenotyped for aggression and anxiety in comparison to BALB/cByJ controls. Microarray data was then used to construct a molecular landscape, based on the mRNAs that were differentially expressed in the brains of BALB/cJ mice. Subsequently, we provided corroborating evidence for the key findings from the landscape through 1H-magnetic resonance imaging and quantitative polymerase chain reactions, specifically in the anterior cingulate cortex (ACC). The molecular landscape predicted that altered GABA signalling may underlie the observed increased aggression and anxiety in BALB/cJ mice. This was supported by a 40% reduction of 1H-MRS GABA levels and a 20-fold increase of the GABA-degrading enzyme Abat in the ventral ACC. As a possible compensation, Kcc2, a potassium-chloride channel involved in GABA-A receptor signalling, was found increased. Moreover, we observed aggressive behaviour that could be linked to altered expression of neuroligin-2, a membrane-bound cell adhesion protein that mediates synaptogenesis of mainly inhibitory synapses. In conclusion, Abat and Kcc2 seem to be involved in modulating aggressive and anxious behaviours observed in BALB/cJ mice through affecting GABA signalling in the ACC.

Impact of Time Delay in Perceptual Decision-Making: Neuronal Population Modeling Approach

Impairments in decision-making are frequently observed in neurodegenerative diseases, but the mechanisms underlying such pathologies remain elusive. In this work, we study, on the basis of novel time-delayed neuronal population model, if the delay in self-inhibition terms can explain those impairments. Analysis of proposed system reveals that there can be up to three positive steady states, with the one having the lowest neuronal activity being always locally stable in nondelayed case. We show, however, that this steady state becomes unstable above a critical delay value for which, in certain parameter ranges, a subcritical Hopf bifurcation occurs. We then apply psychometric function to translate model-predicted ring rates into probabilities that a decision is being made. Using numerical simulations, we demonstrate that for small synaptic delays the decision-making process depends directly on the strength of supplied stimulus and the system correctly identifies to which population the stimulus was applied. However, for delays above the Hopf bifurcation threshold we observe complex impairments in the decision-making process; that is, increasing the strength of the stimulus may lead to the change in the neuronal decision into a wrong one. Furthermore, above critical delay threshold, the system exhibits ambiguity in the decision-making.