Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review

Self-organized Criticality in Neural Networks by Inhibitory and Excitatory Synaptic Plasticity

Neural networks show intrinsic ongoing activity even in the absence of information processing and task-driven activities. This spontaneous activity has been reported to have specific characteristics ranging from scale-free avalanches in microcircuits to the power-law decay of the power spectrum of oscillations in coarse-grained recordings of large populations of neurons. The emergence of scale-free activity and power-law distributions of observables has encouraged researchers to postulate that the neural system is operating near a continuous phase transition. At such a phase transition, changes in control parameters or the strength of the external input lead to a change in the macroscopic behavior of the system. On the other hand, at a critical point due to critical slowing down, the phenomenological mesoscopic modeling of the system becomes realizable. Two distinct types of phase transitions have been suggested as the operating point of the neural system, namely active-inactive and synchronous-asynchronous phase transitions. In contrast to normal phase transitions in which a fine-tuning of the control parameter(s) is required to bring the system to the critical point, neural systems should be supplemented with self-tuning mechanisms that adaptively adjust the system near to the critical point (or critical region) in the phase space. In this work, we introduce a self-organized critical model of the neural network. We consider dynamics of excitatory and inhibitory (EI) sparsely connected populations of spiking leaky integrate neurons with conductance-based synapses. Ignoring inhomogeneities and internal fluctuations, we first analyze the mean-field model. We choose the strength of the external excitatory input and the average strength of excitatory to excitatory synapses as control parameters of the model and analyze the bifurcation diagram of the mean-field equations. We focus on bifurcations at the low firing rate regime in which the quiescent state loses stability due to Saddle-node or Hopf bifurcations. In particular, at the Bogdanov-Takens (BT) bifurcation point which is the intersection of the Hopf bifurcation and Saddle-node bifurcation lines of the 2D dynamical system, the network shows avalanche dynamics with power-law avalanche size and duration distributions. This matches the characteristics of low firing spontaneous activity in the cortex. By linearizing gain functions and excitatory and inhibitory nullclines, we can approximate the location of the BT bifurcation point. This point in the control parameter phase space corresponds to the internal balance of excitation and inhibition and a slight excess of external excitatory input to the excitatory population. Due to the tight balance of average excitation and inhibition currents, the firing of the individual cells is fluctuation-driven. Around the BT point, the spiking of neurons is a Poisson process and the population average membrane potential of neurons is approximately at the middle of the operating interval $[V_{Rest}, V_{th}]$. Moreover, the EI network is close to both oscillatory and active-inactive phase transition regimes. Next, we consider self-tuning of the system at this critical point. The self-organizing parameter in our network is the balance of opposing forces of inhibitory and excitatory populations' activities and the self-organizing mechanisms are long-term synaptic plasticity and short-term depression of the synapses. The former tunes the overall strength of excitatory and inhibitory pathways to be close to a balanced regime of these currents and the latter which is based on the finite amount of resources in brain areas, act as an adaptive mechanism that tunes micro populations of neurons subjected to fluctuating external inputs to attain the balance in a wider range of external input strengths. Using the Poisson firing assumption, we propose a microscopic Markovian model which captures the internal fluctuations in the network due to the finite size and matches the macroscopic mean-field equation by coarse-graining. Near the critical point, a phenomenological mesoscopic model for excitatory and inhibitory fields of activity is possible due to the time scale separation of slowly changing variables and fast degrees of freedom. We will show that the mesoscopic model corresponding to the neural field model near the local Bogdanov-Takens bifurcation point matches Langevin's description of the directed percolation process. Tuning the system at the critical point can be achieved by coupling fast population dynamics with slow adaptive gain and synaptic weight dynamics, which make the system wander around the phase transition point. Therefore, by introducing short-term and long-term synaptic plasticity, we have proposed a self-organized critical stochastic neural field model.:1. Introduction 1.1. Scale-free Spontaneous Activity 1.1.1. Nested Oscillations in the Macro-scale Collective Activity 1.1.2. Up and Down States Transitions 1.1.3. Avalanches in Local Neuronal Populations 1.2. Criticality and Self-organized Criticality in Systems out of Equilibrium 1.2.1. Sandpile Models 1.2.2. Directed Percolation 1.3. Critical Neural Models 1.3.1. Self-Organizing Neural Automata 1.3.2. Criticality in the Mesoscopic Models of Cortical Activity 1.4. Balance of Inhibition and Excitation 1.5. Functional Benefits of Being in the Critical State 1.6. Arguments Against the Critical State of the Brain 1.7. Organization of the Current Work 2. Single Neuron Model 2.1. Impulse Response of the Neuron 2.2. Response of the Neuron to the Constant Input 2.3. Response of the Neuron to the Poisson Input 2.3.1. Potential Distribution of a Neuron Receiving Poisson Input 2.3.2. Firing Rate and Interspike intervals’ CV Near the Threshold 2.3.3. Linear Poisson Neuron Approximation 3. Interconnected Homogeneous Population of Excitatory and Inhibitory Neurons 3.1. Linearized Nullclines and Different Dynamic Regimes 3.2. Logistic Function Approximation of Gain Functions 3.3. Dynamics Near the BT Bifurcation Point 3.4. Avalanches in the Region Close to the BT Point 3.5. Stability Analysis of the Fixed Points in the Linear Regime 3.6. Characteristics of Avalanches 4. Long Term and Short Term Synaptic Plasticity rules Tune the EI Population Close to the BT Bifurcation Point 4.1. Long Term Synaptic Plasticity by STDP Tunes Synaptic Weights Close to the Balanced State 4.2. Short-term plasticity and Up-Down states transition 5. Interconnected network of EI populations: Wilson-Cowan Neural Field Model 6. Stochastic Neural Field 6.1. Finite size fluctuations in a single EI population 6.2. Stochastic Neural Field with a Tuning Mechanism to the Critical State 7. Conclusio

Adaptive dynamical networks

It is a fundamental challenge to understand how the function of a network is related to its structural organization. Adaptive dynamical networks represent a broad class of systems that can change their connectivity over time depending on their dynamical state. The most important feature of such systems is that their function depends on their structure and vice versa. While the properties of static networks have been extensively investigated in the past, the study of adaptive networks is much more challenging. Moreover, adaptive dynamical networks are of tremendous importance for various application fields, in particular, for the models for neuronal synaptic plasticity, adaptive networks in chemical, epidemic, biological, transport, and social systems, to name a few. In this review, we provide a detailed description of adaptive dynamical networks, show their applications in various areas of research, highlight their dynamical features and describe the arising dynamical phenomena, and give an overview of the available mathematical methods developed for understanding adaptive dynamical networks

Perspectives on adaptive dynamical systems

Adaptivity is a dynamical feature that is omnipresent in nature, socio-economics, and technology. For example, adaptive couplings appear in various real-world systems like the power grid, social, and neural networks, and they form the backbone of closed-loop control strategies and machine learning algorithms. In this article, we provide an interdisciplinary perspective on adaptive systems. We reflect on the notion and terminology of adaptivity in different disciplines and discuss which role adaptivity plays for various fields. We highlight common open challenges, and give perspectives on future research directions, looking to inspire interdisciplinary approaches.Comment: 46 pages, 9 figure

Dwelling Quietly in the Rich Club: Brain Network Determinants of Slow Cortical Fluctuations

For more than a century, cerebral cartography has been driven by investigations of structural and morphological properties of the brain across spatial scales and the temporal/functional phenomena that emerge from these underlying features. The next era of brain mapping will be driven by studies that consider both of these components of brain organization simultaneously -- elucidating their interactions and dependencies. Using this guiding principle, we explored the origin of slowly fluctuating patterns of synchronization within the topological core of brain regions known as the rich club, implicated in the regulation of mood and introspection. We find that a constellation of densely interconnected regions that constitute the rich club (including the anterior insula, amygdala, and precuneus) play a central role in promoting a stable, dynamical core of spontaneous activity in the primate cortex. The slow time scales are well matched to the regulation of internal visceral states, corresponding to the somatic correlates of mood and anxiety. In contrast, the topology of the surrounding "feeder" cortical regions show unstable, rapidly fluctuating dynamics likely crucial for fast perceptual processes. We discuss these findings in relation to psychiatric disorders and the future of connectomics.Comment: 35 pages, 6 figure

Self-organization without conservation: Are neuronal avalanches generically critical?

Recent experiments on cortical neural networks have revealed the existence of well-defined avalanches of electrical activity. Such avalanches have been claimed to be generically scale-invariant -- i.e. power-law distributed -- with many exciting implications in Neuroscience. Recently, a self-organized model has been proposed by Levina, Herrmann and Geisel to justify such an empirical finding. Given that (i) neural dynamics is dissipative and (ii) there is a loading mechanism "charging" progressively the background synaptic strength, this model/dynamics is very similar in spirit to forest-fire and earthquake models, archetypical examples of non-conserving self-organization, which have been recently shown to lack true criticality. Here we show that cortical neural networks obeying (i) and (ii) are not generically critical; unless parameters are fine tuned, their dynamics is either sub- or super-critical, even if the pseudo-critical region is relatively broad. This conclusion seems to be in agreement with the most recent experimental observations. The main implication of our work is that, if future experimental research on cortical networks were to support that truly critical avalanches are the norm and not the exception, then one should look for more elaborate (adaptive/evolutionary) explanations, beyond simple self-organization, to account for this.Comment: 28 pages, 11 figures, regular pape

Perspectives on adaptive dynamical systems

Adaptivity is a dynamical feature that is omnipresent in nature, socio-economics, and technology. For example, adaptive couplings appear in various real-world systems, such as the power grid, social, and neural networks, and they form the backbone of closed-loop control strategies and machine learning algorithms. In this article, we provide an interdisciplinary perspective on adaptive systems. We reflect on the notion and terminology of adaptivity in different disciplines and discuss which role adaptivity plays for various fields. We highlight common open challenges and give perspectives on future research directions, looking to inspire interdisciplinary approaches

Slow-wave sleep : generation and propagation of slow waves, role in long-term plasticity and gating

Tableau d’honneur de la Faculté des études supérieures et postdoctorales, 2012-2013.Le sommeil est connu pour réguler plusieurs fonctions importantes pour le cerveau et parmi celles-ci, il y a le blocage de l’information sensorielle par le thalamus et l’amélioration de la consolidation de la mémoire. Le sommeil à ondes lentes, en particulier, est considéré être critique pour ces deux processus. Cependant, leurs mécanismes physiologiques sont inconnus. Aussi, la marque électrophysiologique distinctive du sommeil à ondes lentes est la présence d’ondes lentes de grande amplitude dans le potentiel de champ cortical et l’alternance entre des périodes d’activités synaptiques intenses pendant lesquelles les neurones corticaux sont dépolarisés et déchargent plusieurs potentiels d’action et des périodes silencieuses pendant lesquelles aucune décharge ne survient, les neurones corticaux sont hyperpolarisés et très peu d’activités synaptiques sont observées. Tout d'abord, afin de mieux comprendre les études présentées dans ce manuscrit, une introduction générale couvrant l'architecture du système thalamocortical et ses fonctions est présentée. Celle-ci comprend une description des états de vigilance, suivie d'une description des rythmes présents dans le système thalamocortical au cours du sommeil à ondes lentes, puis par une description des différents mécanismes de plasticité synaptique, et enfin, deux hypothèses sur la façon dont le sommeil peut affecter la consolidation de la mémoire sont présentées. Puis, trois études sont présentées et ont été conçues pour caractériser les propriétés de l'oscillation lente du sommeil à ondes lentes. Dans la première étude (chapitre II), nous avons montré que les périodes d'activité (et de silence) se produisent de façon presque synchrone dans des neurones qui ont jusqu'à 12 mm de distance. Nous avons montré que l'activité était initiée en un point focal et se propageait rapidement à des sites corticaux voisins. Étonnamment, le déclenchement des états silencieux était encore plus synchronisé que le déclenchement des états actifs. L'hypothèse de travail pour la deuxième étude (chapitre III) était que les états actifs sont générés par une sommation de relâches spontanées de médiateurs. Utilisant différents enregistrements à la fois chez des animaux anesthésiés et chez d’autres non-anesthésiés, nous avons montré qu’aucune décharge neuronale ne se produit dans le néocortex pendant les états silencieux du sommeil à ondes lentes, mais certaines activités synaptiques peuvent ii être observées avant le début des états actifs, ce qui était en accord avec notre hypothèse. Nous avons également montré que les neurones de la couche V étaient les premiers à entrer dans l’état actif pour la majorité des cycles, mais ce serait ainsi uniquement pour des raisons probabilistes; ces cellules étant équipées du plus grand nombre de contacts synaptiques parmi les neurones corticaux. Nous avons également montré que le sommeil à ondes lentes et l’anesthésie à la kétamine-xylazine présentent de nombreuses similitudes. Ayant utilisé une combinaison d'enregistrements chez des animaux anesthésiés à la kétamine-xylazine et chez des animaux non-anesthésiés, et parce que l'anesthésie à la kétamine-xylazine est largement utilisée comme un modèle de sommeil à ondes lentes, nous avons effectué des mesures quantitatives des différences entre les deux groupes d'enregistrements (chapitre IV). Nous avons trouvé que l'oscillation lente était beaucoup plus rythmique sous anesthésie et elle était aussi plus cohérente entre des sites d’enregistrements distants en comparaison aux enregistrements de sommeil naturel. Sous anesthésie, les ondes lentes avaient également une amplitude plus grande et une durée plus longue par rapport au sommeil à ondes lentes. Toutefois, les ondes fuseaux (spindles) et gamma étaient également affectées par l'anesthésie. Dans l'étude suivante (Chapitre V), nous avons investigué le rôle du sommeil à ondes lentes dans la formation de la plasticité à long terme dans le système thalamocortical. À l’aide de stimulations pré-thalamiques de la voie somatosensorielle ascendante (fibres du lemnisque médial) chez des animaux non-anesthésiés, nous avons montré que le potentiel évoqué enregistré dans le cortex somatosensoriel était augmenté dans une période d’éveil suivant un épisode de sommeil à ondes lentes par rapport à l’épisode d’éveil précédent et cette augmentation était de longue durée. Nous avons également montré que le sommeil paradoxal ne jouait pas un rôle important dans cette augmentation d'amplitude des réponses évoquées. À l’aide d'enregistrements in vitro en mode cellule-entière, nous avons caractérisé le mécanisme derrière cette augmentation et ce mécanisme est compatible avec la forme classique de potentiation à long terme, car il nécessitait une activation à la fois les récepteurs NMDA et des récepteurs AMPA, ainsi que la présence de calcium dans le neurone post-synaptique. iii La dernière étude incluse dans cette thèse (chapitre VI) a été conçue pour caractériser un possible mécanisme physiologique de blocage sensoriel thalamique survenant pendant le sommeil. Les ondes fuseaux sont caractérisées par la présence de potentiels d’action calcique à seuil bas et le calcium joue un rôle essentiel dans la transmission synaptique. En utilisant plusieurs techniques expérimentales, nous avons vérifié l'hypothèse que ces potentiels d’action calciques pourraient causer un appauvrissement local de calcium dans l'espace extracellulaire ce qui affecterait la transmission synaptique. Nous avons montré que les canaux calciques responsables des potentiels d’action calciques étaient localisés aux synapses et que, de fait, une diminution locale de la concentration extracellulaire de calcium se produit au cours d’un potentiel d’action calcique à seuil bas spontané ou provoqué, ce qui était suffisant pour nuire à la transmission synaptique. Nous concluons que l'oscillation lente est initiée en un point focal et se propage ensuite aux aires corticales voisines de façon presque synchrone, même pour des cellules séparées par jusqu'à 12 mm de distance. Les états actifs de cette oscillation proviennent d’une sommation de relâches spontanées de neuromédiateurs (indépendantes des potentiels d’action) et cette sommation peut survenir dans tous neurones corticaux. Cependant, l’état actif est généré plus souvent dans les neurones pyramidaux de couche V simplement pour des raisons probabilistes. Les deux types d’expériences (kétamine-xylazine et sommeil à ondes lentes) ont montré plusieurs propriétés similaires, mais aussi quelques différences quantitatives. Nous concluons également que l'oscillation lente joue un rôle essentiel dans l'induction de plasticité à long terme qui contribue très probablement à la consolidation de la mémoire. Les ondes fuseaux, un autre type d’ondes présentes pendant le sommeil à ondes lentes, contribuent au blocage thalamique de l'information sensorielle.Sleep is known to mediate several major functions in the brain and among them are the gating of sensory information during sleep and the sleep-related improvement in memory consolidation. Slow-wave sleep in particular is thought to be critical for both of these processes. However, their physiological mechanisms are unknown. Also, the electrophysiological hallmark of slow-wave sleep is the presence of large amplitude slow waves in the cortical local field potential and the alternation of periods of intense synaptic activity in which cortical neurons are depolarized and fire action potentials and periods of silence in which no firing occurs, cortical neurons are hyperpolarized, and very little synaptic activities are observed. First, in order to better understand the studies presented in this manuscript, a general introduction covering the thalamocortical system architecture and function is presented, which includes a description of the states of vigilance, followed by a description of the rhythms present in the thalamocortical system during slow-wave sleep, then by a description of the mechanisms of synaptic plasticity, and finally two hypotheses about how sleep might affect the consolidation of memory are presented. Then, three studies are presented and were designed to characterize the properties of the sleep slow oscillation. In the first study (Chapter II), we showed that periods of activity (and silence) occur almost synchronously in neurons that are separated by up to 12 mm. The activity was initiated in a focal point and rapidly propagated to neighboring sites. Surprisingly, the onsets of silent states were even more synchronous than onsets of active states. The working hypothesis for the second study (Chapter III) was that active states are generated by a summation of spontaneous mediator releases. Using different recordings in both anesthetized and non-anesthetized animals, we showed that no neuronal firing occurs in the neocortex during silent states of slow-wave sleep but some synaptic activities might be observed prior to the onset of active states, which was in agreement with our hypothesis. We also showed that layer V neurons were leading the onset of active states in most of the cycles but this would be due to probabilistic reasons; these cells being equipped with the most numerous synaptic contacts among cortical neurons. We also showed that slow-wave sleep and ketamine-xylazine shares many similarities. v Having used a combination of recordings in ketamine-xylazine anesthetized and non-anesthetized animals, and because ketamine-xylazine anesthesia is extensively used as a model of slow-wave sleep, we made quantitative measurements of the differences between the two groups of recordings (Chapter IV). We found that the slow oscillation was much more rhythmic under anesthesia and it was also more coherent between distant sites as compared to recordings during slow-wave sleep. Under anesthesia, slow waves were also of larger amplitude and had a longer duration as compared to slow-wave sleep. However, spindles and gamma were also affected by the anesthesia. In the following study (Chapter V), we investigated the role of slow-wave sleep in the formation of long-term plasticity in the thalamocortical system. Using pre-thalamic stimulations of the ascending somatosensory pathway (medial lemniscus fibers) in non-anesthetized animals, we showed that evoked potential recorded in the somatosensory cortex were enhanced in a wake period following a slow-wave sleep episode as compared to the previous wake episode and this enhancement was long-lasting. We also showed that rapid eye movement sleep did not play a significant role in this enhancement of response amplitude. Using whole-cell recordings in vitro, we characterized the mechanism behind this enhancement and it was compatible with the classical form of long-term potentiation, because it required an activation of both NMDA and AMPA receptors as well as the presence of calcium in the postsynaptic neuron. The last study included in this thesis (Chapter VI) was designed to characterise a possible physiological mechanism of thalamic sensory gating occurring during sleep. Spindles are characterized by the presence of low-threshold calcium spikes and calcium plays a critical role in the synaptic transmission. Using several experimental techniques, we verified the hypothesis that these calcium spikes would cause a local depletion of calcium in the extracellular space which would impair synaptic transmission. We showed that calcium channels responsible for calcium spikes were co-localized with synapses and that indeed, local extracellular calcium depletion occurred during spontaneous or induced low-threshold calcium spike, which was sufficient to impair synaptic transmission. We conclude that slow oscillation originate at a focal point and then propagate to neighboring cortical areas being almost synchronous even in cells located up to 12 mm vi apart. Active states of this oscillation originate from a summation of spike-independent mediator releases that might occur in any cortical neurons, but happens more often in layer V pyramidal neurons simply due to probabilistic reasons. Both experiments in ketamine-xylazine anesthesia and non-anesthetized animals showed several similar properties, but also some quantitative differences. We also conclude that slow oscillation plays a critical role in the induction of long-term plasticity, which very likely contributes to memory consolidation. Spindles, another oscillation present in slow-wave sleep, contribute to the thalamic gating of information