Statistics of spike trains in conductance-based neural networks: Rigorous results

We consider a conductance based neural network inspired by the generalized Integrate and Fire model introduced by Rudolph and Destexhe. We show the existence and uniqueness of a unique Gibbs distribution characterizing spike train statistics. The corresponding Gibbs potential is explicitly computed. These results hold in presence of a time-dependent stimulus and apply therefore to non-stationary dynamics.Comment: 42 pages, 1 figure, to appear in Journal of Mathematical Neuroscienc

Synchronization in complex networks

Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.Comment: Final version published in Physics Reports. More information available at http://synchronets.googlepages.com

Membrane resonance enables stable and robust gamma oscillations

Neuronal mechanisms underlying beta/gamma oscillations (20-80 Hz) are not completely understood. Here, we show that in vivo beta/gamma oscillations in the cat visual cortex sometimes exhibit remarkably stable frequency even when inputs fluctuate dramatically. Enhanced frequency stability is associated with stronger oscillations measured in individual units and larger power in the local field potential. Simulations of neuronal circuitry demonstrate that membrane properties of inhibitory interneurons strongly determine the characteristics of emergent oscillations. Exploration of networks containing either integrator or resonator inhibitory interneurons revealed that: (i) Resonance, as opposed to integration, promotes robust oscillations with large power and stable frequency via a mechanism called RING (Resonance INduced Gamma); resonance favors synchronization by reducing phase delays between interneurons and imposes bounds on oscillation cycle duration; (ii) Stability of frequency and robustness of the oscillation also depend on the relative timing of excitatory and inhibitory volleys within the oscillation cycle; (iii) RING can reproduce characteristics of both Pyramidal INterneuron Gamma (PING) and INterneuron Gamma (ING), transcending such classifications; (iv) In RING, robust gamma oscillations are promoted by slow but are impaired by fast inputs. Results suggest that interneuronal membrane resonance can be an important ingredient for generation of robust gamma oscillations having stable frequency

On Dynamics of Integrate-and-Fire Neural Networks with Conductance Based Synapses

We present a mathematical analysis of a networks with Integrate-and-Fire neurons and adaptive conductances. Taking into account the realistic fact that the spike time is only known within some \textit{finite} precision, we propose a model where spikes are effective at times multiple of a characteristic time scale $\delta$, where $\delta$ can be \textit{arbitrary} small (in particular, well beyond the numerical precision). We make a complete mathematical characterization of the model-dynamics and obtain the following results. The asymptotic dynamics is composed by finitely many stable periodic orbits, whose number and period can be arbitrary large and can diverge in a region of the synaptic weights space, traditionally called the "edge of chaos", a notion mathematically well defined in the present paper. Furthermore, except at the edge of chaos, there is a one-to-one correspondence between the membrane potential trajectories and the raster plot. This shows that the neural code is entirely "in the spikes" in this case. As a key tool, we introduce an order parameter, easy to compute numerically, and closely related to a natural notion of entropy, providing a relevant characterization of the computational capabilities of the network. This allows us to compare the computational capabilities of leaky and Integrate-and-Fire models and conductance based models. The present study considers networks with constant input, and without time-dependent plasticity, but the framework has been designed for both extensions.Comment: 36 pages, 9 figure

Chaotic exploration and learning of locomotion behaviours

We present a general and fully dynamic neural system, which exploits intrinsic chaotic dynamics, for the real-time goal-directed exploration and learning of the possible locomotion patterns of an articulated robot of an arbitrary morphology in an unknown environment. The controller is modeled as a network of neural oscillators that are initially coupled only through physical embodiment, and goal-directed exploration of coordinated motor patterns is achieved by chaotic search using adaptive bifurcation. The phase space of the indirectly coupled neural-body-environment system contains multiple transient or permanent self-organized dynamics, each of which is a candidate for a locomotion behavior. The adaptive bifurcation enables the system orbit to wander through various phase-coordinated states, using its intrinsic chaotic dynamics as a driving force, and stabilizes on to one of the states matching the given goal criteria. In order to improve the sustainability of useful transient patterns, sensory homeostasis has been introduced, which results in an increased diversity of motor outputs, thus achieving multiscale exploration. A rhythmic pattern discovered by this process is memorized and sustained by changing the wiring between initially disconnected oscillators using an adaptive synchronization method. Our results show that the novel neurorobotic system is able to create and learn multiple locomotion behaviors for a wide range of body configurations and physical environments and can readapt in realtime after sustaining damage

Memory capacity in the hippocampus

Neural assemblies in hippocampus encode positions. During rest, the hippocam- pus replays sequences of neural activity seen during awake behavior. This replay is linked to memory consolidation and mental exploration of the environment. Re- current networks can be used to model the replay of sequential activity. Multiple sequences can be stored in the synaptic connections. To achieve a high mem- ory capacity, recurrent networks require a pattern separation mechanism. Such a mechanism is global remapping, observed in place cell populations. A place cell fires at a particular position of an environment and is silent elsewhere. Multiple place cells usually cover an environment with their firing fields. Small changes in the environment or context of a behavioral task can cause global remapping, i.e. profound changes in place cell firing fields. Global remapping causes some cells to cease firing, other silent cells to gain a place field, and other place cells to move their firing field and change their peak firing rate. The effect is strong enough to make global remapping a viable pattern separation mechanism. We model two mechanisms that improve the memory capacity of recurrent net- works. The effect of inhibition on replay in a recurrent network is modeled using binary neurons and binary synapses. A mean field approximation is used to de- termine the optimal parameters for the inhibitory neuron population. Numerical simulations of the full model were carried out to verify the predictions of the mean field model. A second model analyzes a hypothesized global remapping mecha- nism, in which grid cell firing is used as feed forward input to place cells. Grid cells have multiple firing fields in the same environment, arranged in a hexagonal grid. Grid cells can be used in a model as feed forward inputs to place cells to produce place fields. In these grid-to-place cell models, shifts in the grid cell firing patterns cause remapping in the place cell population. We analyze the capacity of such a system to create sets of separated patterns, i.e. how many different spatial codes can be generated. The limiting factor are the synapses connecting grid cells to place cells. To assess their capacity, we produce different place codes in place and grid cell populations, by shuffling place field positions and shifting grid fields of grid cells. Then we use Hebbian learning to increase the synaptic weights be- tween grid and place cells for each set of grid and place code. The capacity limit is reached when synaptic interference makes it impossible to produce a place code with sufficient spatial acuity from grid cell firing. Additionally, it is desired to also maintain the place fields compact, or sparse if seen from a coding standpoint. Of course, as more environments are stored, the sparseness is lost. Interestingly, place cells lose the sparseness of their firing fields much earlier than their spatial acuity. For the sequence replay model we are able to increase capacity in a simulated recurrent network by including an inhibitory population. We show that even in this more complicated case, capacity is improved. We observe oscillations in the average activity of both excitatory and inhibitory neuron populations. The oscillations get stronger at the capacity limit. In addition, at the capacity limit, rather than observing a sudden failure of replay, we find sequences are replayed transiently for a couple of time steps before failing. Analyzing the remapping model, we find that, as we store more spatial codes in the synapses, first the sparseness of place fields is lost. Only later do we observe a decay in spatial acuity of the code. We found two ways to maintain sparse place fields while achieving a high capacity: inhibition between place cells, and partitioning the place cell population so that learning affects only a small fraction of them in each environment. We present scaling predictions that suggest that hundreds of thousands of spatial codes can be produced by this pattern separation mechanism. The effect inhibition has on the replay model is two-fold. Capacity is increased, and the graceful transition from full replay to failure allows for higher capacities when using short sequences. Additional mechanisms not explored in this model could be at work to concatenate these short sequences, or could perform more complex operations on them. The interplay of excitatory and inhibitory populations gives rise to oscillations, which are strongest at the capacity limit. The oscillation draws a picture of how a memory mechanism can cause hippocampal oscillations as observed in experiments. In the remapping model we showed that sparseness of place cell firing is constraining the capacity of this pattern separation mechanism. Grid codes outperform place codes regarding spatial acuity, as shown in Mathis et al. (2012). Our model shows that the grid-to-place transformation is not harnessing the full spatial information from the grid code in order to maintain sparse place fields. This suggests that the two codes are independent, and communication between the areas might be mostly for synchronization. High spatial acuity seems to be a specialization of the grid code, while the place code is more suitable for memory tasks. In a detailed model of hippocampal replay we show that feedback inhibition can increase the number of sequences that can be replayed. The effect of inhibition on capacity is determined using a meanfield model, and the results are verified with numerical simulations of the full network. Transient replay is found at the capacity limit, accompanied by oscillations that resemble sharp wave ripples in hippocampus. In a second model Hippocampal replay of neuronal activity is linked to memory consolidation and mental exploration. Furthermore, replay is a potential neural correlate of episodic memory. To model hippocampal sequence replay, recurrent neural networks are used. Memory capacity of such networks is of great interest to determine their biological feasibility. And additionally, any mechanism that improves capacity has explanatory power. We investigate two such mechanisms. The first mechanism to improve capacity is global, unspecific feedback inhibition for the recurrent network. In a simplified meanfield model we show that capacity is indeed improved. The second mechanism that increases memory capacity is pattern separation. In the spatial context of hippocampal place cell firing, global remapping is one way to achieve pattern separation. Changes in the environment or context of a task cause global remapping. During global remapping, place cell firing changes in unpredictable ways: cells shift their place fields, or fully cease firing, and formerly silent cells acquire place fields. Global remapping can be triggered by subtle changes in grid cells that give feed-forward inputs to hippocampal place cells. We investigate the capacity of the underlying synaptic connections, defined as the number of different environments that can be represented at a given spatial acuity. We find two essential conditions to achieve a high capacity and sparse place fields: inhibition between place cells, and partitioning the place cell population so that learning affects only a small fraction of them in each environments. We also find that sparsity of place fields is the constraining factor of the model rather than spatial acuity. Since the hippocampal place code is sparse, we conclude that the hippocampus does not fully harness the spatial information available in the grid code. The two codes of space might thus serve different purposes