From Synaptic Interactions to Collective Dynamics in Random Neuronal Networks Models: Critical Role of Eigenvectors and Transient Behavior.

The study of neuronal interactions is at the center of several big collaborative neuroscience projects (including the Human Connectome Project, the Blue Brain Project, and the Brainome) that attempt to obtain a detailed map of the entire brain. Under certain constraints, mathematical theory can advance predictions of the expected neural dynamics based solely on the statistical properties of the synaptic interaction matrix. This work explores the application of free random variables to the study of large synaptic interaction matrices. Besides recovering in a straightforward way known results on eigenspectra in types of models of neural networks proposed by Rajan and Abbott (2006), we extend them to heavy-tailed distributions of interactions. More important, we analytically derive the behavior of eigenvector overlaps, which determine the stability of the spectra. We observe that on imposing the neuronal excitation/inhibition balance, despite the eigenvalues remaining unchanged, their stability dramatically decreases due to the strong nonorthogonality of associated eigenvectors. This leads us to the conclusion that understanding the temporal evolution of asymmetric neural networks requires considering the entangled dynamics of both eigenvectors and eigenvalues, which might bear consequences for learning and memory processes in these models. Considering the success of free random variables theory in a wide variety of disciplines, we hope that the results presented here foster the additional application of these ideas in the area of brain sciences.

Unraveling the fluctuations of animal motor activity.

Human and animal behavior exhibits power law correlations whose origin is controversial. In this work, the spontaneous motion of laboratory rodents was recorded during several days. It is found that animal motion is scale-free and that the scaling is introduced by the inactivity pauses both by its length as well as by its specific ordering. Furthermore, the scaling is also demonstrable in the rates of event's occurrence. A comparison with related results in humans is made and candidate models are discussed to provide clues for the origin of such dynamics.

Flattened cortical maps of cerebral function in the rat: a region-of-interest approach to data sampling, analysis and display.

We describe a method for the measurement, analysis and display of cerebral cortical data obtained from coronal brain sections of the adult rat. In this method, regions-of-interest (ROI) are selected in the cortical mantle in a semiautomated fashion using a radial grid overlay, spaced in 15 degrees intervals from the midline. ROI measurements of intensity are mapped on a flattened two-dimensional surface. Topographic maps of statistical significance at each ROI allow for the rapid viewing of group differences. Cortical z-scores are displayed with the boundaries of brain regions defined according to a standard atlas of the rat brain. This method and accompanying software implementation (Matlab, Labview) allow for compact data display in a variety of autoradiographic and histologic studies of the structure and function of the rat brain.

Brain activity for spontaneous pain of postherpetic neuralgia and its modulation by lidocaine patch therapy.

Postherpetic neuralgia (PHN) is a debilitating chronic pain condition, yet there is a lack of knowledge regarding underlying brain activity. Here we identify brain regions involved in spontaneous pain of PHN (n=11) and determine its modulation with Lidoderm therapy (patches of 5% lidocaine applied to the PHN affected body part). Continuous ratings of fluctuations of spontaneous pain during fMRI were contrasted to ratings of fluctuations of a bar observed during scanning, at three sessions: (1) pre-treatment baseline, (2) after 6h of Lidoderm treatment, and (3) after 2 weeks of Lidoderm use. Overall brain activity for spontaneous pain of PHN involved affective and sensory-discriminative areas: thalamus, primary and secondary somatosensory, insula and anterior cingulate cortices, as well as areas involved in emotion, hedonics, reward, and punishment: ventral striatum, amygdala, orbital frontal cortex, and ventral tegmental area. Generally, these activations decreased at sessions 2 and 3, except right anterior insular activity which increased with treatment. The sensory and affective activations only responded to the short-term treatment (6h of Lidoderm); while the ventral striatum and amygdala (reward-related regions) decreased mainly with longer-term treatment (2 weeks of Lidoderm). Pain properties: average magnitude of spontaneous pain, and responses on Neuropathic Pain Scale (NPS), decreased with treatment. The ventral striatal and amygdala activity best reflected changes in NPS, which was modulated only with longer-term treatment. The results show a specific brain activity pattern for PHN spontaneous pain, and implicate areas involved in emotions and reward as best reflecting changes in pain with treatment.

Single subject pharmacological-MRI (phMRI) study: modulation of brain activity of psoriatic arthritis pain by cyclooxygenase-2 inhibitor.

We use fMRI to examine brain activity for pain elicited by palpating joints in a single patient suffering from psoriatic arthritis. Changes in these responses are documented when the patient ingested a single dose of a selective cyclooxygenase-2 inhibitor (COX-2i). We show that mechanical stimulation of the painful joints exhibited a cortical activity pattern similar to that reported for acute pain, with activity primarily localized to the thalamus, insular, primary and secondary somatosensory cortices and the mid anterior cingulum. COX-2i resulted in significant decreased in reported pain intensity and in brain activity after 1 hour of administration. The anterior insula and SII correlated with pain intensity, however no central activation site for the drug was detected. We demonstrate the similarity of the activation pattern for palpating painful joints to brain activity in normal subjects in response to thermal painful stimuli, by performing a spatial conjunction analysis between these maps, where overlap is observed in the insula, thalamus, secondary somatosensory cortex, and anterior cingulate. The results demonstrate that one can study effects of pharmacological manipulations in a single subject where the brain activity for a clinical condition is delineated and its modulation by COX-2i demonstrated. This approach may have diagnostic and prognostic utility.

Heart rate dynamics in monoamine oxidase-A- and -B-deficient mice.

Heart rate (HR) dynamics were investigated in mice deficient in monoamine oxidase A and B, whose phenotype includes elevated tissue levels of norepinephrine, serotonin, dopamine, and phenylethylamine. In their home cages, spectral analysis of R-R intervals revealed more pronounced fluctuations at all frequencies in the mutants compared with wild-type controls, with a particular enhancement at 1-4 Hz. No significant genotypic differences in HR variability (HRV) or entropies calculated from Poincaré plots of the R-R intervals were noted. During exposure to the stress of a novel environment, HR increased and HRV decreased in both genotypes. However, mutants, unlike controls, demonstrated a rapid return to baseline HR during the 10-min exposure. Such modulation may result from an enhanced vagal tone, as suggested by the observation that mutants responded to cholinergic blockade with a decrease in HRV and a prolonged tachycardia greater than controls. Monoamine oxidase-deficient mice may represent a useful experimental model for studying compensatory mechanisms responsible for changes in HR dynamics in chronic states of high sympathetic tone.

Increased baroreceptor response in mice deficient in monoamine oxidase A and B.

The recent development of mice doubly deficient for monoamine oxidase A and B (MAO-A/B, respectively) has raised questions about the impact of these mutations on cardiovascular function, in so far as these animals demonstrate increased tissue levels of the vasoactive amines serotonin, norepinephrine, dopamine, and phenylethylamine. We recorded femoral arterial pressures and electrocardiograms in adult MAO-A/B-deficient mice during halothane-nitrous oxide anesthesia as well as 30 min postoperatively. During both anesthesia and recovery, systolic, diastolic, and mean arterial pressures were 10-15 mmHg lower in MAO-A/B-deficient mice compared with normal controls (P < 0.01). Mutants also showed a greater baroreceptor-mediated reduction in heart rate in response to hypertension after intravenous pulses of phenylephrine or angiotensin II. Tachycardia elicited in response to hypotension after nitroprusside was greater in mutants than in controls. Heart rate responsiveness to changes in arterial pressure was abolished after administration of glycopyrrolate, with no differences in this phenomenon noted between genotypes. These data suggest that prevention of hypertension may occur in chronic states of catecholaminergic/indoleaminergic excess by increased gain of the baroreflex.

Adaptive learning by extremal dynamics and negative feedback.

We describe a mechanism for biological learning and adaptation based on two simple principles: (i) Neuronal activity propagates only through the network's strongest synaptic connections (extremal dynamics), and (ii) the strengths of active synapses are reduced if mistakes are made, otherwise no changes occur (negative feedback). The balancing of those two tendencies typically shapes a synaptic landscape with configurations which are barely stable, and therefore highly flexible. This allows for swift adaptation to new situations. Recollection of past successes is achieved by punishing synapses which have once participated in activity associated with successful outputs much less than neurons that have never been successful. Despite its simplicity, the model can readily learn to solve complicated nonlinear tasks, even in the presence of noise. In particular, the learning time for the benchmark parity problem scales algebraically with the problem size N, with an exponent k approximately 1.4.

Noise-induced memory in extended excitable systems.

We describe a form of memory exhibited by extended excitable systems driven by stochastic fluctuations. Under such conditions, the system self-organizes into a state characterized by power-law correlations, thus retaining long-term memory of previous states. The exponents are robust and model independent. We discuss implications of these results for the functioning of cortical neurons as well as for networks of neurons.

Noise in neurons is message dependent.

Neuronal responses are conspicuously variable. We focus on one particular aspect of that variability: the precision of action potential timing. We show that for common models of noisy spike generation, elementary considerations imply that such variability is a function of the input, and can be made arbitrarily large or small by a suitable choice of inputs. Our considerations are expected to extend to virtually any mechanism of spike generation, and we illustrate them with data from the visual pathway. Thus, a simplification usually made in the application of information theory to neural processing is violated: noise is not independent of the message. However, we also show the existence of error-correcting topologies, which can achieve better timing reliability than their components.

Learning from mistakes.

We re-examine the commonly held view that learning and memory necessarily require potentiation of synapses. A simple neuronal model of self-organized learning with no positive reinforcement is presented. The strongest synapses are selected for propagation of activity. Active synaptic connections are temporarily "tagged" and subsequently depressed if the resulting output turns out to be unsuccessful. Thus, all learning occurs by mistakes. The model operates at a highly adaptive state with low activity. Previously stored patterns may be swiftly retrieved when the environment and the demands of the brain change. The combined process of: (i) activity selection by extremal "winner-take-all" dynamics; and (ii) the subsequent weeding out of synapses may be viewed as synaptic Darwinism. We argue that all the features of the model are biologically plausible and discuss our results in light of recent experiments by Fitzsimonds et al. on back-propagation of long-term depression, by Xu et al. on facilitation of long-term depression in the hippocampus by behavioural stress, and by Frey and Morris on synaptic tagging.

Noise-induced tuning curve changes in mechanoreceptors.

Fibers from the tibial nerve of rat were isolated and spike activity recorded using monopolar hook electrodes. The receptive field (RF) of each recorded unit on the glabrous skin of the foot was mechanically stimulated with waveforms comprised of various frequency sine waves in addition to increasing levels of white noise. Single-unit responses were recorded for both rapidly adapting (RA) and slowly adapting (SA) units. Signal-to-noise ratio (SNR) of the output was quantified by the correlation coefficient (C1) between the input sine wave and the nerve responses. The addition of noise enhanced signal transmission in both RA and SA fibers. With increasing noise, the initially inverted "V"-shaped, zero-noise tuning curves for RA fibers broadened and eventually inverted. There was a large expansion of the frequencies that the RA receptor responded to with increasing noise input. On the other hand, the typical shape of the SA fiber tuning curves remained invariant, at all noise levels tested. C1 values continued to increase with larger noise input for higher frequencies, but did not do so at the lowest frequencies. For both RA and SA fibers the responses with added noise tended to be rate modulated at the low-frequency end, and followed nonlinear stochastic resonance (SR) properties at the higher frequencies. The changes in the tuning properties due to noise found here, as well as preliminary psychophysics data, imply that external noise is relevant for sensing small periodic signals in the environment. All current models of sensory perception assume that the tuning properties of receptors determined in the absence of noise are preserved during everyday tasks. Our results indicate that this is not true in a noisy environment.

General relation between variance-time curve and power spectral density for point processes exhibiting 1/f beta-fluctuations, with special reference to heart rate variability.

Counting statistics in the form of the variance-time curve provides an alternative to spectral analysis for point processes exhibiting 1/f beta-fluctuations, such as the heart beat. However, this is true only for beta < 1. Here, the case of general beta is considered. To that end, the mathematical relation between the variance-time curve and power spectral density in the presence of 1/f beta-noise is worked out in detail. A modified version of the variance-time curve is presented, which allows us to deal also with the case beta > or = 1. Some applications to the analysis of heart rate variability are given.

Nonlinear dynamics of rate-dependent activation in models of single cardiac cells.

Recent studies in isolated cardiac tissue preparations have demonstrated the applicability of a one-dimensional difference equation model describing the global behavior of a driven nonpacemaker cell to the understanding of rate-dependent cardiac excitation. As a first approximation to providing an ionic basis to complex excitation patterns in cardiac cells, we have compared the predictions of the one-dimensional model with those of numerical simulations using a modified high-dimensional ionic model of the space-clamped myocyte. Stimulus-response ratios were recorded at various stimulus magnitudes, durations, and frequencies. Iteration of the difference equation model reproduced all important features of the ionic model results, including a wide spectrum of stimulus-response locking patterns, period doubling, and irregular (chaotic) dynamics. In addition, in the parameter plane, both models predict that the bifurcation structure of the cardiac cell must change as a function of stimulus duration, because stimulus duration modifies the type of supernormal excitability present at short diastolic interval. We conclude that, to a large extent, the bifurcation structure of the ionic model under repetitive stimulation can be understood by two functions: excitability and action potential duration. The characteristics of these functions depend on the stimulus duration.

Sustained vortex-like waves in normal isolated ventricular muscle.

Sustained reentrant excitation may be initiated in small (20 x 20 x less than 0.6 mm) preparations of normal ventricular muscle. A single appropriately timed premature electrical stimulus applied perpendicularly to the wake of a propagating quasiplanar wavefront gives rise to circulation of self-sustaining excitation waves, which pivot at high frequency (5-7 Hz) around a relatively small "phaseless" region. Such a region develops only very low amplitude depolarizations. Once initiated, most episodes of reentrant activity last indefinitely but can be interrupted by the application of an appropriately timed electrical stimulus. The entire course of the electrical activity is visualized with high temporal and spatial resolution, as well as high signal-to-noise ratio, using voltage-sensitive dyes and optical mapping. Two- and three-dimensional graphics of the fluorescence changes recorded by a 10 x 10 photodiode array from a surface of 12 x 12 mm provide sequential images (every msec) of voltage distribution during a reentrant vortex. The results suggest that two-dimensional vortex-like reentry in cardiac muscle is analogous to spiral waves in other biological and chemical excitable media.

Directional differences in excitability and margin of safety for propagation in sheep ventricular epicardial muscle.

Computer simulations and isolated tissue experiments were used to characterize the relation between excitability and margin of safety for propagation in anisotropic ventricular myocardium. Longitudinal, uniform transverse, and nonuniform transverse tissue directions were modeled in a one-dimensional Beeler-Reuter based cable. Stimulation threshold was smallest in the nonuniform transverse direction. The safety factor for propagation was determined in the model as the total axial charge that was available for depolarizing downstream tissue divided by the threshold charge that was just sufficient for continued propagation and was largest in the longitudinal direction. The strength-interval plot for the junction between simulated longitudinal and nonuniform transverse directions identified a range of stimulus strengths and intervals that resulted in nonuniform transverse but not longitudinal propagation. When high values of transverse resistance were used, higher stimulus strengths during premature stimulation resulted in longitudinal but not nonuniform transverse propagation. The experimental strength interval plots from 17 L-shaped preparations of isolated sheep epicardial muscles had similar characteristics. In nine additional L-shaped tissue experiments, changing extracellular K+ concentrations from 4 to 20 mM resulted in progressive membrane depolarization and conduction impairment in both directions. However, in eight of nine experiments, complete block occurred first in the transverse direction. In one experiment, block was simultaneous in both directions. We conclude that, under normal conditions, threshold requirements for active propagation are lower for transverse than for longitudinal propagation. In addition, when active membrane properties are impaired, the safety factor for propagation is larger in the direction along the longitudinal axis of the cells.

Low dimensional chaos in cardiac tissue.

Chaos is a term used to characterize aperiodic activity arising in a dynamical system, or in a set of equations describing the system's temporal evolution as a result of a deterministic mechanism that has sensitive dependence on initial conditions. Chaos, in that sense, has been proposed to make an important contribution to normal and abnormal cardiac rhythms. To date, however, descriptions of chaos in heart tissue have been limited primarily to periodically forced cardiac pacemakers. Because many cardiac rhythm disturbances, particularly those initiated or perpetuated by re-entrant excitation, originate from within non-pacemaker cardiac tissues, demonstrations of chaos in non-pacemaker tissue might provide a deterministic explanation for a wide variety of complex dysrhythmias. Here we report experimental evidence for chaotic patterns of activation and action potential characteristics in externally driven, non-spontaneously active Purkinje fibres and ventricular muscle. The results indicate that there is an apparent link between the mechanism of low dimensional chaos and the occurrence of reflected responses which could lead to more spatially disorganized phenomena. A detailed mechanism for the low dimensional chaos observed experimentally is pursued using a difference equation model. Critical features of the model include a non-monotonic relationship between recovery time during rhythmic stimulation and the state of membrane properties, and a steeply sloped recovery of membrane properties over certain ranges of recovery times. Besides explaining our results, the analytical model may pertain to irregular dynamics in other excitable systems, particularly the intact dysrhythmic heart.

Supernormal excitability as a mechanism of chaotic dynamics of activation in cardiac Purkinje fibers.

Supernormality, which can be defined as greater than normal excitability during or immediately after action potential repolarization, has been observed in a variety of cardiac preparations. However, as yet, no description of the dynamics of tissue response to repetitive stimulation in the presence of supernormal or relatively supernormal excitability has appeared. Isolated sheep cardiac Purkinje fibers (2-5 mm in length) were superfused with Tyrode's solution and stimulated with depolarizing current pulses through a suction pipette. Recovery of excitability, restitution of the action potential duration, and response patterns were measured in each fiber for a wide range of current amplitudes and stimulation frequencies. When the potassium chloride concentration of the Tyrode's solution was decreased from 7 to 4 mM, the excitability recovery function consistently changed from monophasic ("normal") to triphasic ("supernormal'). During repetitive stimulation at increasing rates, normal preparations responded only with gradual changes in the activation ratio, expressed as periodic phase-locked responses (i.e., Wenckebach-like patterns, etc.). Supernormal preparations showed a nonmonotonic change in the activation ratio, as well as complex aperiodic response patterns. Numerical results from an analytical model gave a quantitative basis for the relation between nonmonotonicity in the excitability function and the development of complex rhythms in cardiac Purkinje fibers. Both our experimental and theoretical results indicate that the presence of supernormality and the slope of the action potential duration restitution curve at short diastolic intervals are responsible for the development of chaotic dynamics. Moreover, our results give an accurate description of the supernormality phenomenon, predict the behavior expected under such conditions, and provide insight about the role of membrane recovery in determining regular and irregular frequency-dependent rhythm and conduction disturbances.