Ionic concentrations fluctuate significantly during seizures. Substantial increase of extracellular K+ is found during electrically- or pharmacologically-induced paroxysmal activity, along with increase of intracellular Na+. These changes of the ionic concentrations trigger various homeostasis mechanisms such as glial uptake and Na+/K+ ATPase. While Na+/K+ ATPase is one of the most studied proteins, its role in epilepsy remains unclear. Using computational model of in vivo epileptiform activity, we found that increase of intracellular Na+ during epileptiforms leads to significant activation of Na+/K+ ATPase; this increase mediates hyperpolarizing current by Na+/K+ pump that contributes to termination of seizure and postictal depression state. Deficiencies of the Na+/K+ ATPase promote continuous epileptiform activity. In terms of dynamics, the mechanism underlying the smooth transition is due to a safe bifurcation of a homoclinic orbit of a saddle-node equilibrium state terminating the quiescence period of bursting. Overall, our study demonstrated a complex role played by Na+/K+ ATPase in developing of epileptiform activity and may suggest new targets for antiepileptic drugs.
In Parkinson's disease, increased power of oscillations in firing rate has been observed throughout the cortico-basal-ganglia circuit. In particular, the excessive oscillations in the beta range (13-30Hz) have been shown to be associated with difficulty of movement initiation. However, on the basis of experimental data alone it is difficult to determine where these oscillations are generated, due to complex and recurrent structure of the cortico-basal-ganglia-thalamic circuit. This talk will describe a computational model of a subset of basal-ganglia that is able to reproduce experimentally observed patterns of activity. The analysis of the model suggests where and under which conditions the beta oscillations are produced.
Transient dynamics is pervasive in the human brain and poses challenging problems both in mathematical tractability and clinical observability. We investigate statistical properties of transient cortical wave patterns with characteristic forms (shape, size, duration) in a canonical reaction-diffusion model with mean field inhibition. The patterns are formed by a ghost near a saddle-node bifurcation in which a stable traveling wave (node) collides with its critical nucleation mass (saddle). Similar patterns have been observed with fMRI in migraine. Our results support the controversial idea that waves of cortical spreading depression (SD) have a causal relationship with the headache phase in migraine. We suggest a congruence between the prevalence of two subtypes, migraine without aura and migraine with aura, and the statistical properties of the traveling waves. We briefly discuss model-based control and means by which neuromodulation techniques may affect pathways of pain formation.
A major challenge for the brain is to maintain stability while retaining sufficient flexibility to grow and experience plasticity. The operating state must be such that excessive excitation or insufficient excitation does not result in response to external stimuli. How these opposing constraints reconcile is currently not well understood. We show that after synaptic potentiation, a network of in vitro hippocampal neurons returns to a homeostatic state after widespread increases in firing.
This work describes a framework for creating patient-specific neural mass models using intracranial electroencephalogram (iEEG) recordings from patients with epilepsy. Neural mass models are used in epilepsy research to relate physiological parameters to iEEG in an attempt to generate hypotheses about the generation and termination of seizures. We will fuse the data and the neural mass model to estimate parameters that specify the shape of post-synaptic potentials, connectivity strength between neural populations, and firing thresholds. A nonlinear version of the Kalman Filter is used with an augmented state-space model to solve the estimation problem. Results from artificial data and patient data show that this is a promising framework.
We study a data-driven model of thalamocortical (TC) relay neuron to examine the TC relay responses to an excitatory input train, under inhibitory signals. We first incorporate recording data as inhibitory signals to the TC model to investigate the mechanism underlying deep brain stimulation (DBS) which has been proven clinically effective to relieve motor symptoms for Parkinsonian patients. Then we explore the closed-loop stimulation paradigm using a parkinsonian network model of the basal-ganglia thalamocortical circuit. Our computational results show that the type of stimulation, based on a filtered version of the local field potential, significantly improves the fidelity of thalamocortical (TC) relay. To further understand the different scenarios of TC relay responses, we analyze the entrainment of the TC neuron to periodic signals that alternate between 'on' and 'off', respectively. By exploiting invariant sets of the system and their associated invariant fiber bundles that foliate the phase space, we reduce the 3D TC model to a 2D map. Based on this map, we reproduce the possible scenarios of TC relay responses observed in the data-driven model.
Several authors have discussed previously the use of loglinear models, often called maximum entropy models, for analyzing spike train data to detect synchrony. The usual loglinear modeling techniques, however, do not allow for time-varying firing rates that typically appear in stimulus-driven (or action-driven) neurons, nor do they incorporate non-Poisson history effects or covariate effects. I will outline a generalization of the usual approach, which combines point process regression models of individual-neuron activity with loglinear models of multiway synchronous interaction (Kass, Kelly, and Loh, 2011, Annals of Applied Statistics; Kelly and Kass, 2012, Neural Computation). I will also describe a method, based on Bayesian control of false discoveries, for assessing the large number of pairs of neurons that are typically examined in a single experiment. Preliminary physiological results come from Utah array recordings in V1.
A seizure represents an extreme deviation from normal brain activity. In this talk, we will consider some characteristics of the seizure as observed across spatial and temporal scales in human patients. We will focus specifically on changes in the rhythmic voltage activity, and consider techniques to characterize these changes. We will also discuss a mathematical model consistent with the stereotyped dynamics observed at seizure termination, and use this model to propose what happens dynamically when a seizure fails to self- terminate.
Prominent beta frequency oscillations appear in the basal ganglia of Parkinsons disease patients. The dynamical mechanisms by which these beta oscillations arise are unknown. Using mathematical models, we show that robust beta frequency rhythms can emerge from inhibitory interactions between striatal medium spiny neurons. Our modeling studies propose that the pathologic beta oscillations in Parkinsons disease may arise as an indirect effect of striatal dopamine loss on the striatal cholinergic system. Experimental testing of our model by infusion of the cholinergic agonist carbachol into normal, mouse striatum induced pronounced, reversible beta oscillations in the striatal local field potential. These results suggest the prominent beta oscillations in Parkinsons disease may bethe result of an exaggeration of normal striatal network dynamics.
Deep brain stimulation (DBS) has emerged as an exciting new possibility for the treatment of neuropsychiatric disorders. Theoretical and experimental evidence suggest that when DBS is applied to neural tissue, it responds with the generation of action potentials in axons. This is especially pertinent to neuropsychiatric DBS applications because they are currently focused on stimulation of sub-cortical white matter. Unfortunately, it is unclear which specific pathways within the white matter are responsible for generating therapeutic benefit from the stimulation. We are attempting to identify those pathways via creation of patient-specific computational models that combine diffusion-tensor tractography, axonal stimulation predictions, and clinical outcome analyses. Our early results from patients with treatment refractory depression suggest that pathways associated with the ventral medial pre-frontal cortex and accumbens play an important role in therapeutic benefit. Identification of specific target pathways for modulation provides opportunities to improve clinical selection of electrode placement and stimulation settings for DBS devices.
Despite decades of research, the biochemical and neurophysiological causes of depression remain unknown. Furthermore, although selective serotonin reuptake inhibitors (SSRIs) block the reuptake of serotonin and alleviate depression in some patients, it is not clear how or why they work. Mathematical models of serotonin synthesis, release, and reuptake can shed light on the control mechanisms of the serotonin system and suggest hypotheses about the action of SSRIs. We will discuss two of the standard hypotheses and propose a new hypothesis.
Parkinsonʼs disease has been traditionally thought of as a dopaminergic disease in which cells of the substantia nigra pars compacta (SNc) die. However, accumulating evidence implies an important role for the serotonergic system in Parkinsonʼs disease in general and in physiological responses to levodopa therapy, the first line of treatment. We use a mathematical model to investigate the consequences of levodopa therapy on the serotonergic system and on the pulsatile release of dopamine (DA) from dopaminergic and serotonergic terminals in the striatum.
We will also ask, and propose an answer to, the question of what serotonin is doing in the striatum anyway?
Motor symptoms of Parkinson's disease have been associated with the synchronized oscillatory activity in the cortico-basal ganglia-thalamic circuits. Here we will present our observations of the patterns of synchronized activity obtained through simultaneous intraoperative recordings of spikes and LFP in the basal ganglia and cortical EEG in parkinsonian patients. We discuss the temporal patterning of the observed synchronized patterning. We show how the synchronization of EEG in motor and prefrontal areas (which can be obtained noninvasively) is predictive of the spike-LFP synchrony in subthalamic nucleus. We also consider the observed phenomena within the framework of mathematical models of cortico-basal ganglia circuits.
A mathematical model that integrates the dynamics of cell membrane potential, ion homeostasis, cell volume, mitochondrial ATP production, mitochondrial and ER Ca2+ handling, IP3 production and GTP-binding protein coupled receptor signaling is considered. Simulations with the model support recent experimental data showing a protective effect of stimulating astrocytic P2Y1 receptors following cerebral ischemic stroke. The model is analyzed in order to better understand the mathematical behavior of the equations and to provide insights into the underlying biological data. This approach yields explicit formulas determining how changes in IP3-mediated calcium release, under varying conditions of oxygen and the energy substrate pyruvate, affect mitochondrial ATP production, and is utilized to predict rate-limiting variables in P2Y1 receptors enhanced astrocyte protection after cerebral ischemic stroke. This is joint work with C. Diekman, C. Fall and J. Lechleiter.
Epilepsy is a disorder typically characterized by synchronous neuronal activity. While many studies have focused on understanding macro level network activity using electrophysiological approaches, less is known about the underlying spatiotemporal dynamics at the micro level of individual cells. Here, we use two-photon calcium imaging to study the functional network structure of individual neurons in the dentate gyrus of chronically epileptic networks. Under hyperexcitable conditions, slices from the epileptic dentate gyrus display recurrent synchronous network level activity. However, using a functional clustering activity to examine the contributions of individual neurons to these events, we show that these macro level synchronizations are actually composed of the co-activation of subsets of spatially localized neuronal clusters. Furthermore, different combinations of clusters are active during each event, revealing that although the events appear similar when viewed at the macro level of network synchronization, they are in fact quite variable and display little repetition in the micro level structure of events.
Schizophrenia is a highly complex disorder with deficits in various modalities, including perceptual deficits in the auditory domain. One useful approach for characterizing the neurophysiological basis of auditory cortical deficits has been the utilization of EEG steady-state responses entrained to external periodic stimuli, allowing testing of the functional state of the networks subserving synchronous activity. Individuals with schizophrenia show deficits in entraining to auditory stimuli in the gamma-frequency range, particularly at 40 Hz. Employing the auditory steady-state paradigm, we hypothesized that administration of amphetamine, which increases dopamine levels by facilitating release and inhibiting reuptake, would improve gamma-range auditory entrainment in schizophrenia patients and produce minimal change or decrease it for healthy controls. Our empirical results supported our hypothesis and showed that this effect was more specific to the gamma-range and not reproduced for other frequencies. To verify and replicate this observation computationally, a network composed of 50 excitatory (Wang-Buzsaki) and 20 fast-spiking inhibitory (Wang-Buzsaki) neurons coupled with all-to-all connectivity was formulated. Periodic forcing at different frequencies (40, 30, and 20 Hz) was applied and the network entrainment to the driving frequencies was analyzed. The network’s behavior corresponding to different levels of dopamine, implemented as varying levels of the leak K+ conductance of the inhibitory neurons, was analyzed. For the 40 Hz forcing case, parametrically varying the K+ leak conductance revealed an inverted-U shaped relationship, with low gamma band power at both low and high conductance levels, with optimal synchronization occurring at intermediate conductance levels. A similar inverted-U shaped relationship was obtained for the periodic forcing at 30 Hz, though less robustly. However, 20 Hz forcing case did not reveal such a non-monotonic relationship, suggesting that the effect was more specific to periodic forcing at gamma band range, in line with the empirical findings in auditory steady-state entrainment in healthy controls and individuals with schizophrenia. In conclusion, our results show that the physiological effects of dopamine on single fast-spiking interneurons can give rise to a non-monotonic relationship between cortical gamma band synchrony and dopamine levels with absence of a similar relationship for entrained responses at lower frequency ranges (e.g. beta), consistent with our empirical findings in schizophrenia of specific modulations oscillations in the gamma band by dopamine modulation.
A procedure is developed for finding an energy-optimal stimulus which gives a positive Lyapunov exponent, and hence desynchronization, for a neural population. Not only does it achieve desynchronization, but it also does so using less energy than recently proposed methods, suggesting a powerful alternative to pulsatile stimuli for deep brain stimulation. Furthermore, we calculate error bounds on the optimal stimulus which will guarantee a minimum Lyapunov exponent. Also, a related control strategy is developed for desynchronizing neurons based on the population's phase distribution.