Search

Bernstein Conference 2022 abstract. http://bernstein-conference.de

Bernstein Conference 2022 abstract. http://bernstein-conference.de

Bernstein Conference 2022 abstract. http://bernstein-conference.de

Bernstein Conference 2022 abstract. http://bernstein-conference.de

The activity of spiking network models exhibits fast oscillations (>200 Hz), caused by inhibition-dominated excitatory-inhibitory loops [1, 2]. As correlations between pairs of neurons are weak in nature and models, fast oscillations have so far received little attention.Today’s models of cortical networks with natural numbers of neurons and synapses [3] remove any uncertainty about down-scaling artifacts [4]. Fast oscillations here arise as vertical stripes in raster diagrams. We discuss experimental detectability of oscillations, ask whether they are an artifact of simplified models, and identify adaptations to control them.The population rate spectrum decomposes into single-neuron power spectra (∼N) and cross-spectra of pairs of neurons (∼N2) [5,6]. For low numbers of neurons (100) and weak correlations, the single-neuron spectra dominate the compound spectrum. Coherent oscillations in the population activity may thus go unnoticed in experimental spike recordings. Population measures obtained from large neuron ensembles (e.g., LFP), however, should show a pronounced peak.Cortical network models allow an investigation from different angles. We rule out artifacts of time-discrete simulation and investigate the effect of distributed synaptic delays: exponential distributions decrease the oscillation amplitude, expected by their equivalence to low-pass filtering [7], whereas truncated Gaussian distributions are ineffective.Surprisingly, a model of V1 [8], with the same architecture, but fewer synapses per neuron, does not exhibit fast oscillations. Mean-field theory shows that loops within each inhibitory population cause fast oscillations. Peak frequency and amplitude are determined by eigenvalues of the effective connectivity matrix approaching instability [9]. Reducing the connection density decreases the eigenvalues, increasing their distance to instability; we thus expect weaker oscillations.Counter to expectation and simulation, mean-field theory predicts an increase, explained by an overestimation of the transfer function at high frequencies [10]: the initial network appears to be linearly unstable, with |λ|>1; reduced connectivity seemingly destabilizes the system. A semi-analytical correction restores qualitative agreement with simulation.The work points at the importance of models with realistic cell densities and connectivity, and illustrates the productive interplay of simulation-driven and analytical approaches.References 1. Brunel, N. Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. JComputNeurosci 8, 183–208 (2000)., 10.1371/journal.pcbi.1006359 2. Brunel, N. & Wang, X.-J. What Determines the Frequency of Fast Network Oscillations With Irregular Neural Discharges? I. Synaptic Dynamics and Excitation-Inhibition Balance. JNeurophysiol 90, 415–430 (2003)., 10.1152/jn.01095.2002 3. Potjans, T. C. & Diesmann, M. The Cell-Type Specific Cortical Microcircuit: Relating Structure and Activity in a Full-Scale Spiking Network Model. CerebCortex 24, 785–806 (2014)., 10.1093/cercor/bhs358 4. van Albada, S. J., Helias, M. & Diesmann, M. Scalability of asynchronous networks is limited by one-to-one mapping between effective connectivity and correlations. ploscb 11, e1004490 (2015)., 10.1371/journal.pcbi.1004490 5. Harris, K. D., & Thiele, A. Cortical state and attention. Nature Reviews Neuroscience, 12(9), 509-523 (2011)., 10.1038/nrn3084 6. Tetzlaff, T., Helias, M., Einevoll, G. T., & Diesmann, M. Decorrelation of neural-network activity by inhibitory feedback. PLoS Comput Biol, 8(8), e1002596 (2012)., 10.1371/journal.pcbi.1002596 7. Mattia, M., Biggio, M., Galluzzi, A. & Storace, M. Dimensional reduction in networks of non-Markovian spiking neurons: Equivalence of synaptic filtering and heterogeneous propagation delays. PLoS Comput Biol 15, e1007404 (2019)., 10.1371/journal.pcbi.1007404 8. Schmidt, M. et al. A multi-scale layer-resolved spiking network model of resting-state dynamics in macaque visual cortical areas. ploscb 14, e1006359 (2018)., 10.1023/a:1008925309027 9. Bos, H., Diesmann, M. & Helias, M. Identifying Anatomical Origins of Coexisting Oscillations in the Cortical Microcircuit. ploscb 12, e1005132 (2016)., 10.1371/journal.pcbi.1005132 10. Schuecker, J., Diesmann, M. & Helias, M. Modulated escape from a metastable state driven by colored noise. Phys. Rev. E (2015)., 10.1103/PhysRevE.92.052119

Frontiers in neuroinformatics 14, 12 (2020). doi:10.3389/fninf.2020.00012, Published by Frontiers Research Foundation, Lausanne

Experimental findings suggest that cortical networks operate in a balanced state [1] in which strong recurrent inhibition suppresses single cell input correlations [2,3]. The balanced state, however, only restricts the average correlations in the network, the distribution of correlations between individual neurons is not constrained. We here investigate this distribution and establish a functional relation between the dynamical state of the system and the variance of correlations as a function of cortical distance. The former is characterized by the spectral radius, a measure for how strong a signal is damped while traversing the network. To this end, we develop a theory that captures the heterogeneity of correlations across neurons. Technically, we derive a mean-field theory that assumes the distribution of correlations to be self-averaging; i.e. the same in any realization of the random network. This is possible by taking advantage of the symmetry of the disorder-averaged [4] effective connectivity matrix. We here demonstrate that spatially organized, balanced network models predict rich pairwise correlation structures with spatial extent far beyond the range of direct connections [5]. Massively parallel spike recordings of macaque motor cortex quantitatively confirm this prediction. We show that the range of these correlations depends on the spectral radius, which offers a potential dynamical mechanism to control the spatial range on which neurons cooperatively perform computations.

Many phenomena observed in biological neural networks can only be explained by assuming nonlinearinteractions. Due to effects like synaptic failure and channel noise, neuronal dynamics is also inherentlystochastic. The investigation of the interaction of both of these properties is challenging because due to thenonlinearity, correlations of higher order influence those of lower order. Nonlinear, stochastic systems exhibita plethora of dynamical states. The cortex, especially, is often suggested to operate close to a critical pointat which linear response theory fails since the neural dynamics is dominated by large fluctuations on alllength scales [1].This is the realm of the Renormalization Group (RG), stemming from statistical and particle physics. Weuse this technique in the form introduced by Kenneth G. Wilson [2] to study a two-dimensional stochasticneural field model. Its connectivity is composed of two Gaussian kernels, one mimicking the excitatoryand the other the inhibitory input. Its dynamics is given by a nonlinear gain and intrinsic nonlinear neurondynamics. Gaussian white noise accounting for unspecified external input and intrinsic stochasticity drivesour system. In its long-distance approximation, this model is similar to that proposed by Kardar, Parisi, andZhang (KPZ) [3].Along the lines taken in their approach, we derive RG-flow equations describing the couplings in ourneural field model on different length scales. From this, one finds the upper critical dimension dc=2,which corresponds to the dimension of networks in cortex. Above dc , mean-field theory is exact as theGaussian fixed point is attractive for small interactions, whereas below dc , the interaction dominates thebehavior. For d=dc , however, we find that the Gaussian fixed point becomes unstable and the interactionparameter flows into a strong coupling regime – similar to the KPZ model. A strong coupling fixed pointmay be present, which would indicate self-similarity - the signature of a critical state. Our analysis thereforeimplies certain constrains on the architecture of the neural network (within our model) if it is supposed towork at a critical point. For example, we conclude that we get stable dynamics only if the excitatoryinputs extend wider than the inhibitory ones.AcknowledgementsPartly supported by seed funds MSCALE and CLS002 of the RWTH University; the JARA Center for Doctoral studies within the graduate School for Simulation and Data Science (SSD); the Helmholtz association: Young investigator's grant VH-NG-1028; EU-Grant 785907 (HBP).References[1] Beggs, J. M., Plenz, D. (2003), Neuronal avalanches in neocortical circuits, Journal of Neuroscience, 23, 35, 11167–11177., 10.1523/JNEUROSCI.23-35-11167.2003[2] Wilson, K. G., Kogut, J. (1974), The renormalization group and the ε expansion, Physics Reports, 12, 2 , 75 - 199.[3] Kardar, M., Parisi, G., Zhang, Y.(1986), Dynamic Scaling of Growing Interfaces, Phys. Rev. Lett. 56, 889–892., 10.1103/PhysRevLett.56.88

Frontiers in neuroinformatics 12, 2 (2018). doi:10.3389/fninf.2018.00002, Published by Frontiers Research Foundation, Lausanne

Originally invented to describe magnetism, the Ising model has proven to be useful in many other applications, as, for example, inference problems in computer science, socioeconomic physics, the analysis of neural data [1,2,3] and modeling of neural networks (binary neurons). Despite its simplicity, there exists no general solution to the Ising model, i.e. the partition function is unknown in the case of an interacting system. Mean field theory is often used as an approximation being exact in the noninteracting case and for infinite dimensions. A correction term to the mean field approximation of Gibb's free energy (the effective action) of the Ising model was given by Thouless, Anderson and Palmer (TAP) [4] as a “fait accompli” and was later derived by different methods in [5,6,7], where also higher order terms were computed.We present a diagrammatic derivation (Feynman diagrams) of these correction terms and embed the problem in the language of field theory. Furthermore, we show how the iterative construction of the effective action used in the Ising case generalizes to arbitrary non-Gaussian theories.References[1] Tkacik, G., Schneidman, E., Berry II, M. J., Bialek, W. (2008): Ising models for networks of real neurons. arXiv:q-bio/0611072[2] Roudi, Y., Tyrcha, J. and Hertz, J.A. (2009): Ising model for neural data: Model quality and approximate methods for extracting functional connectivity. Phys. Rev. E 79, 051915[3] Hertz, J.A., Roudi, Y. and Tyrcha, J (2011): Ising models for inferring network structure from spike data. arXiv:1106.1752.[4] Thouless, D.J., Anderson, P.W. and Palmer, R.G. (1977): Solution of ’Solvable model of a spin glass’. Phil. Mag. 35 3, 593 – 601[5] Georges, A. and Yedidia, J.S. (1991): How to expand around mean-field theory using high-temperature expansions. J. Phys. A 24, 2173 – 2192[6] Parisi, G. and Potters, M. (1995): Mean-Field equations for spin models with orthogonal interaction matrices. J. Phys. A 28, 5267 – 5285[7] Tanaka, T. (2000): Information Geometry of Mean-Field Approximation. Neur. Comp. 12, 1951-1968.Acknowledgements. This work was partially supported by HGF young investigator’s group VH-NG-1028, Helmholtz portfolio theme SMHB, Juelich Aachen Research Alliance (JARA), and EU Grant 604102 (Human Brain Project, HBP).

Despite the large amount of shared input between nearby neurons in cortical circuits, massively parallel spiking recordings of various in vivo networks exhibit pairwise covariances in ensembles of neuronal spike trains that are on average close to zero [1]. The low average has been well understood in terms of active decorrelation by inhibitory feedback [2,3] in networks that operate far away from the critical point, which marks the onset of avalanche-like activity [4]. Experiments, however, also show large variability of covariances across pairs of neurons. An explanation for their wide distribution in relation to the static (quenched) disorder of the connectivity in recurrent networks is so far elusive.Here we combine ideas from spin-glass theory [5] with a generating function representation for the joint probability distribution of the network activity [6] to derive a finite-size mean-field theory that reduces a disordered to a highly symmetric network with fluctuating auxiliary fields (Fig. 1). The theory relates the statistics of covariances to the statistics of connections, in particular the largest eigenvalue of the connectivity matrix, and explains the experimentally observed covariance distributions [7]. The analytical expressions expose that both, average and dispersion of the latter, diverge at a critical point which has been studied in terms of a transition from regular to chaotic dynamics [8,9,10]. This critical point does not arise from net excitation, but rather from disorder in networks with balanced excitation and inhibition. Applying these results to recordings from motor cortex suggests its operation close to this breakdown of linear stability. References:1. Ecker AS, Berens P, Keliris GA, Bethge M, Logothetis NK: Decorrelated Neuronal Firing in Cortical Microcircuits. Science 2010, 327:584-587.2. Renart A, De La Rocha J, Bartho P, Hollender L, Parga N, Reyes A, Harris KD: The asynchronous State in Cortical Circuits. Science 2010, 327:587-590.3. Tetzlaff T, Helias M, Einevoll G, Diesmann M: Decorrelation of neural-network activity by inhibitory feedback. PLOS Comput. Biol. 2010, 8(8):e1002596.4. Beggs JM, Plenz D: Neuronal avalanches in neocortical circuits. J. Neurosci. 2003, 23:11167-11177.5. Sompolinsky H, Zippelius A: Relaxational dynamics of the Edwards-Anderson model and the mean-field theory of spin-glasses. Phys. Rev. B 1982, 25:6860-6875.6. Chow C, Buice M: Path Integral Methods for Stochastic Differential Equations. J Math Neurosci. 2015, 5:8. 7. Dahmen D, Diesmann M, Helias M: Distributions of covariances as a window into the operational regime of neuronal networks. arXiv 2016, 1605.04153 [cond-mat.dis-nn].8. Sompolinsky H, Crisanti A, Sommers HJ: Chaos in Random Neural Networks, Phys. Rev. Lett. 1988, 61:259-262.

Two important parts of electrophysiological recordings are the spike times and the local field potential (LFP), which is considered to primarily reflect input activity. In [1], it was shown by unitary event analysis [2,3] that excess synchronous spike events are locked to the phase of LFP beta-oscillations more strongly than spikes not part of such events. Denker et al. proved by a statistical model that this finding could be explained by the existence of cell assemblies, i.e. groups of (excitatory) neurons that are more strongly connected amongst each other than to the rest of the network.To study the influence of the LFP on the correlated single neuron activities first for a simple model capturing the main properties of cortical neural networks, we examine a balanced network of homogeneously connected binary model neurons [4] receiving input from a sinusoidal perturbation [5]. The Glauber dynamics of the network is simulated and approximated by mean-field theory. Treating the periodic input in linear response theory, the cyclostationary first two moments are analytically computed, which agree with their simulated counterparts over a wide parameter range. The deviations of the zero-time lag correlations from their stationary values consist of two summands owing to the modulated susceptibility (one via direct modulation, one via modulated mean activity) and one to the driving of the autocorrelations. For some parameters, this leads to resonant correlations and non-resonant mean activities. Our results can help to answer the question how oscillations in mesoscopic signals and spike correlations interact. As a next step, our model could be extended to include cell assemblies [6], which will allow us to compare our results with the experimental findings more closely.figure caption:A: Contributions to the time-dependent variation of the correlations in linear perturbation theory. B: The deviation of the correlations from their stationary value is maximal for a certain frequency even for this setting with a connectivity matrix having solely purely real eigenvalues. References:[1] Denker M., Cerebral Cortex, 21:2681--2695, 2011, The Local Field Potential Reflects Surplus Spike Synchrony. [2] Grün S, Diesmann M, Aertsen A. Neural Comput., 14:43--80, 2002a, Unitary events in multiple single-neuron spiking activity: I. detection and significance. [3] Grün S, Diesmann M, Aertsen A. Neural Comput., 14:81--119, 2002b, Unitary events in multiple single-neuron spiking activity: II. Nonstationary data. [4] Ginzburg I., Sompolinsky H., Phys. Rev. E 50(4):3171--3191, 1994, Theory of correlations in stochastic neural networks.[5] Kühn T., Helias M., arXiv:1607.08552, 2016, Correlated activity of periodically driven binary networks.[6] Litwin-Kumar A., & Doiron B., Nature Neur., 15(11):1498--1505, 2012, Slow dynamics and high variability in balanced cortical networks with clustered connections.

One of the major challenges of computational neuroscience is the integration of available multi-scale experimental data into coherent models of the brain. Building a link between structure and activity, such a model would serve as a tool to identify the mechanisms giving rise to experimental observations. However, even if a model integrates experimental data on brain anatomy to the best of our knowledge, it will be underconstrained. Therefore, physiological observations should guide the exploration of the model's parameter ranges within the uncertainty of the anatomical data. To this end, we here use a mean-field reduction of spiking network dynamics to investigate the mechanisms that give rise to stable and physiologically realistic activity, which often remain unclear from simulation results alone. This reduction allows us to include activity constraints to shape the phase space of large-scale network models. In particular, we apply the theory to a spiking multi-area model of macaque visual cortex (Schmidt et al., 2013) and increase its fixed-point firing rates to a realistic level while preserving the basin of attraction of this fixed point. To achieve this, we control the location of the separatrix dividing the phase space into realistic low-activity behavior and unrealistic high-activity states (Fig. 1). We identify components on the macroscopic level of cortical areas and on the layer-specific population level that are critical for the stability properties of the network. In particular, we find a subcircuit of two frontal areas, 46 and FEF, which crucially influence the network stability. Moreover, we identify connections to the excitatory population of layer 5 to be critical for the network dynamics, in line with experimental findings (Sanchez-Vives et al., 2000; Beltramo et al., 2013). By systematically refining these connections to a small degree, we obtain realistic layer- and area-specific firing rates in the visual cortex model.

Brain-scale networks exhibit a breathtaking heterogeneity in the dynamical properties and parameters of their constituents. At cellular resolution, the entities of theory are neurons and synapses and over the past decade researchers have learned to manage the heterogeneity of neurons and synapses with efficient data structures. Already early parallel simulation codes stored synapses in a distributed fashion such that a synapse solely consumes memory on the compute node harboring the target neuron. As petaflop computers with some 100,000 nodes become increasingly available for neuroscience, new challenges arise for neuronal network simulation software: Each neuron contacts on the order of 10,000 other neurons and thus has targets only on a fraction of all compute nodes; furthermore, for any given source neuron, at most a single synapse is typically created on any compute node. From the viewpoint of an individual compute node, the heterogeneity in the synaptic target lists thus collapses along two dimensions: the dimension of the types of synapses and the dimension of the number of synapses of a given type. Here we present a data structure taking advantage of this double collapse using metaprogramming techniques. After introducing the relevant scaling scenario for brain-scale simulations, we quantitatively discuss the performance on two supercomputers. We show that the novel architecture scales to the largest petascale supercomputers available today.

Fast oscillations of the population firing rate in the gamma range (50-200 Hz), where each individual neuron fires irregularly with a low rate, have been observed in networks of simulated leaky integrate-and-fire (LIF) neurons as well as in population signals in the living brain.An analytical explanation of oscillatory population rates has been given in [1]. However a systematic approach identifying circuits responsible for specific oscillations in a potentially complicated structured network of populations of neurons is currently not available. Such a method would provide a tool for the identification of connections responsible for the emergence and propagation of oscillations in anatomically constrained networks and shed light on local dynamical mechanisms amplifying or suppressing certain frequencies . It could also be used to design a LIF-network with desired spectral features.In this study we consider the spectra of population firing rates produced by networks with population specific connectivity. The populations are composed of randomly connected LIF-neurons. In our analysis we make use of the formalism developed in [2], who recently closed the gap between the descriptions of linear rate models and LIF-neurons. We derive an expression for the effective connectivity matrix which incorporates the anatomical (synaptic weights, in-degrees) as well as the dynamical properties [3] of the circuit. We are able to predict the spectra of the population firing rates for any connectivity pattern that allows for asynchronous activity. The analytically obtained predictions are validated by comparison with simulation results of a multi-layered cortical network model [4].Decomposing the effective connectivity matrix at a frequency where the spectrum shows a peak reveals the dominating eigenmode. By reconstructing the anatomical and dynamical contributions to this eigenmode we can narrow down the physical connections associated with the peak in the power spectrum.