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Spike Train Correlations Induced by Anatomical Microstructure Stefan Rotter Computational Neuroscience Lab Bernstein Center Freiburg & Faculty of Biology University of Freiburg, Germany NeuroSeeker MathStatNeuro workshop • Laboratoire J. A. Dieudonné, Nice • September 8 – 10, 2015 Please note: This document contains both unpublished and copyrighted material. FOR PERSONAL USE ONLY! No part may be reproduced, copied or used for any purpose without the written permission of authors or copyright owners. [email protected] Dynamic neuronal networks of the brain Correlations and population signals Disentangling multi-synaptic pathways Inferring connectivity from correlations Dynamic neuronal networks of the brain Correlations and population signals Disentangling multi-synaptic pathways Inferring connectivity from correlations s e ps na sy dedrite cell body axon Ramón y Cajal, 1900 Braitenberg, 1978 The leaky integrate-and-fire model C U̇ + 1 U − Urest = I R dynamical variables: X X U (t) membrane potential I(t) input current fixed parameters: C membrane capacitance R membrane resistance Urest Uthresh resting potential threshold potential Simplifications implied by the leaky integrate-and-fire model: point neuron linearity of integration time-invariance all parts of the neuron are iso-potential linear differential equation / linear system parameters do not change in time Current injection into neurons Chemical synapses Source: Mark Bear, Barry Connors, Michael Paradiso Neuroscience: Exploring the Brain, Third Edition, 2006 Intracellular recording in vivo 20 −50 10 −52 −54 −56 −10 membrane potential (mV) membrane potential (mV) 0 −20 −30 −40 −58 −60 −62 −64 −50 −66 −60 −70 −68 0 500 1000 time (ms) 1500 2000 2500 −70 0 50 100 150 time (ms) 200 250 300 courtesy of V. Bringuier and Y. Fregnac The leaky integrate-and-fire neuron model 25 λE = 400 spikes/s λI = 200 spikes/s 100 200 A = 2.7 mV τm = 20 ms Membrane potential (mV) 20 15 10 5 0 −5 −10 0 50 I(t) = 150 X k 250 Time (ms) Jk δ(t − tk ) 300 350 400 (here: Jk = ±A) 450 500 The Mouse Cortex 2×87 mm3 Total volume Total number of neurons 16 000 000 Number of sensory input fibres < 1 000 000 Length of axonal tree 10–40 mm Length of dendritic tree 4 mm Range of axons 1/0.2 mm Range of dendrites 0.2 mm 90 000 /mm3 Density of neurons Density of axons 4 km/mm3 Density of dendrites 0.4 km/mm3 Density of synapses 700 000 000 /mm3 Synapses per neuron 8 000 Probability of synaptic contact Relative density of axons Relative density of dendrites 0.1 10 −5 /10−3 10−3 Valentino Braitenberg & Almut Schüz Cortex: Statistics and Geometry of Neuronal Connectivity Second Edition, Berlin: Springer, 1998 Biological neuronal networks (BNN) excitatory neuron inhibitory neuron background neuron High-Performance Neuro-Computing (HPNC) compute cluster Hathor # processors/cores RAM connectivity 24 × 2 × 2 = 96 24 × 8 GByte = 192 GByte high-speed Infiniband http://www.nest-initiative.org Dynamic neuronal networks of the brain Correlations and population signals Disentangling multi-synaptic pathways Inferring connectivity from correlations Two different network topologies Biological neurons Hybrid neurons Mean input to individual neurons is identical in both cases! Population fluctuations depend on correlations 20 20 10 20 Biological pre neurons D 100 1000 exc 500 50 0 100 200 2000 Neuron ID 1500 20 inh 1500 Counts Neuron ID 2000 Counts C 10 Hybridpre neurons 100 1000 500 50 0 0 100 Time (ms) Kriener et al., Neural Computation, 2008 200 0 Time (ms) Population fluctuations depend on correlations 20 20 10 20 Biological pre neurons D 100 1000 exc 500 50 0 100 200 2000 Neuron ID 1500 20 inh 1500 Counts Neuron ID 2000 Counts C 10 Hybridpre neurons 100 1000 500 50 0 0 100 200 0 Time (ms) Time (ms) X X X Var Xi = Var Xi + Cov Xi , Xj i Kriener et al., Neural Computation, 2008 i i6=j Shared input structure differs A C B Dale: −1 10 D Hybrid: k k l l 0 10 E −2 10 up http://www.nature.com/natureneuroscience Recording from localized neuronal populations Buzsáki, Nature Neuroscience, 2004 Figure 1 Unit isolation quality varies as a function of distance from the electrode. Multisite electrodes (a wire tetrode, for example) can estimate the position of the recorded neurons by triangulation. trode/o lighted Because invasive neuron Further ative ac gives r require Because majorit comput the neo to hypo mini- a blobs), of prin types. T the sim neuron to the h The extrace matical rons7,17 neuron Tuning of the LFP during arm movements 3 4 2 5 1 6 8 7 Contralateral movement LFP (µV) 8 7 6 5 4 3 2 1 750 100 50 0 −500 0 Time (ms) 750 Mehring et al., Nature Neuroscience, 2003 20 20 ... Decoding of arm movements 0.5 0.8 P 0.4 0.6 0.4 0 Dist.−prox. (mm) d 50 0.2 0 0.6 LFP LFP 0.3 SUA MUA 0.2 LFP&SUA e LFP&MUAcc=(0.74,0.83) 0.1 Chance level 0 10 20 30 40 0 Number of electrodes Mehring et al., Nature Neuroscience, 2003 −50 −50 0 Lateral (mm) 50 0 LFP cc=(0.74,0.83) e 0 −50 −50 50 0 50 Dist.−prox. (mm) b 1 Dist.−prox. (mm) a Dist.−prox. (mm) d 50 SUA 0 Lateral (mm) c 0 −50 −50 Sim.rec. Diff.days f 1 cc=(0.93,0.84) Chance level 50 0. 0. 0 0.8 0.6 cc 0.4 0.2 0 Lateral (mm) 0 0 50 2 4 6 8 Number of electrodes −50 −50 SUA 0 Po 0. 0 Dynamic neuronal networks of the brain Correlations and population signals Disentangling multi-synaptic pathways Inferring connectivity from correlations Neuronal microcircuits Convergent synaptic input onto inhibitory interneurons. dimensional rendering of axonal contacts onto a postsynaptic neuron. s at the top represent cell bodies of neurons within the functionally ane. Axons of a horizontally tuned neuron (cell 4; green) and a tuned neuron (cell 10; red) descend and make synapses (small yellow o dendrites of an inhibitory interneuron (cyan). The axonal and segments leading to the convergence were independently traced by a 180 270 90 Direction (degrees) 360 synaptic potentials (EPSPs), in current–clamp mode (Fig. 1 C and D). In 270 experiments, we took measurements from 1,345 neurons and 3,446 pairs of neurons. To ensure statistical ro- second person, blind to the original segmentation (thick tracing). Cell bodies and axons coloured by orientation preference, as in Fig. 1b. Scale bar, 50 mm. b, c, Electron micrographs showing the synapses onto the inhibitory neuron from cell 4 (b) and cell 10 (c) with corresponding colours overlaid. Scale bar, 1 mm. d, e, Orientation tuning curves derived from in vivo calcium imaging of the cell bodies of cell 4 (d) and cell 10 (e). Coloured bars and arrows, stimulus orientation and direction. DF/F, change in fluorescence. Error bars, 6s.e.m. E B A 0.2 0.1 λ: -5 mm L: 2 mm a Excitatory target Inhibitory target c F C 30 um 2 1 11 3 9 4 8 7 d 0.3 0 100 200 300 0 100 200 300 6 0.1 G 10 100 µm 0.0 0.2 12 b 0.3 Reciprocal Non-Reciprocal Overall (p) Connection Probability Connection Probability Connection Probability 0 5 0.0 0.3 0.2 0.1 0.0 0 100 200 300 Intersomatic Distance [um] 12 13 5 11 12 13 11 an ab th ne tie bi ex ne re a fir fo (P iso nu ta te pr bu D 5 9 9 te re th 3A sy ce ev te 10 2 8 2 10 8 6 6 3 4 14 3 1 7 7 Characterized pyramidal neuron Synapse 4 Dendritic fragment Cell body in EM volume From anatomy to connectivity graphs. a, Three-dimensional (dendritic fragments) are drawn as squares. (From top to bottom Fig. and left 1. toPair-wise connectivity. (A) Morphological staining of a cluster of 12 of the dendrites, axons and cell bodies of 14 neurons in the right: functionally characterized cells 5, 2, 7; 13, 6, 14; 1; 10; 11, 3; 9; 12, 4; and 8.) cells recorded simultaneously. (B) Region of the somatosensory cortex where lly imaged plane (coloured according to their orientation preference, c, Three-dimensional rendering of the arbors and cell bodies of functionally recordings were carried out. (C) Connectivity diagram of neurons in D. (D) as in Fig. 1b), and the dendrites and cell bodies of all their characterized neurons, along with postsynaptic targets that either receive Example ptic targets traced in the EM volume (magenta, excitatory targets; convergent input from multiple functionally characterized neurons, or were of recorded traces in an experimental session. A different neuron is Mouse cortex Rat somatosensory remaining neurons werecortex recorded bitory targets; spines on visual postsynaptic targets not shown; themselves functionally characterized (Supplementary Movie 5). d,stimulated A subset of and the responses of the ntary MovieCalcium 5). Scale bar, 100 mm. b, Directed of the the network graph showing only the connections in c, all independently verified in columns). [Scale bars: (displayed horizontal, 100 recording ms; vertical, 1 in mVvitro (15 mV imaging innetwork vivo diagram + electron microscopy in vitro 12-fold patch lly characterized cells and their targets, derived from a. Postsynaptic (from top to bottom and left to right: functionally characterized cells 5, action 2, 7; 13, potentials)]. (E–G) Connection probability profiles as a function of for et al., Perin et al., PNAS, 2011 (magenta) Bock and inhibitory (cyan)Nature, targets with 2011 cell bodies contained 6; 10; 11, 3; 12, 9, 8 and 4). distance. Error bars represent SEM. e EM volume are drawn as circles. Other postsynaptic targets AT U R E | VO L 4 7 1 | 1 0 M A R C H 2 0 1 1 ©2011 Macmillan Publishers Limited. All rights reserved 2 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1016051108 ro se 20 nu cr so Sm in by ap m sh nu Sy How does connectivity induce correlations. . . Pernice, Staude, Cardanobile, Rotter, PLoS Computational Biology 7(5): e1002059, 2011 Pernice, Staude, Cardanobile, Rotter, Phys Rev E 85: 031916, 2012 Pernice, Rotter, Journal of Statistical Mechanics P03008, 2013 Rangan, PRL 102(15): 158101, 2009; PRE 80(3): 036101, 2009 Trousdale, Hu, Shea-Brown, Josić, PLoS Computational Biology 8(3): e1002408, 2012 . . . and to which degree can connectivity be inferred from correlations? Undirected and directed graphs 3 3 2 2 4 4 1 5 1 5 7 6 7 6 A graph consists of vertices (nodes) and edges (links). Each edge connects one pair of vertices. Connections can be either undirected (left) or directed (right). The adjacency matrix of an undirected graph 3 2 4 1 5 0 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 1 0 7 6 The adjacency matrix A = (aij ) fully describes the graph, provided that each vertex has been assigned a unique label. We have aij = 1 if vertex i and vertex j are connected, and aij = 0 otherwise. Because edges are undirected, the matrix is symmetric, i.e. aij = aji for all i and j, and A = AT . The adjacency matrix of a directed graph 3 2 4 1 5 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 1 1 0 7 6 Again, the adjacency matrix A = (aij ) fully describes a graph with labeled vertices. We set aij = 1 if there is a link from j to i, and aij = 0 otherwise. Because here edges are directed, the matrix is asymmetric, i.e. aij 6= aji and A 6= AT . The transposed matrix AT corresponds to a graph with all arrows reversed. The in-degree and out-degree of a directed graph 3 2 4 1 5 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 1 1 0 7 6 The in-degree of a vertex is the number of its incoming edges, the number of its outgoing edges is called out-degree. In the adjacency matrix, the in-degree is the sum of all entries in the corresponding row. The out-degree is the sum of all entries in the corresponding column. Graphs with weighted or multiple edges excitatory neuron inhibitory neuron background neuron Paths in a graph 3 2 3 2 4 4 1 1 5 5 7 7 6 6 Starting from a given node, one can follow the links (if there are any) through the graph, respecting their orientation. If the outdegree of a vertex is larger than one, the path bifurcates. The graph that indicates which nodes can be reached after two hops can have multiple edges between any two vertices (not shown). Powers of the adjacency matrix 3 2 0 0 0 0 0 1 0 0 1 0 1 1 2 0 0 0 0 0 1 0 1 0 1 0 1 2 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 1 2 4 1 5 7 6 The graph that indicates which vertex can be reached after two hops corresponds to the square of the adjacency matrix A2 . Because this derived graph can have multiple edges between any two vertices, A2 can have entries greater than one. Accordingly, the matrix power Ak corresponds to paths of length k. The Hawkes process A network of N nodes (neurons) is described in terms of spike train firing rate external input interaction kernel si (t) = X Hawkesmo mo Hawkes δ(t − tik ) k yi (t) = si (t) Spi Spik wh wher Rat Rate y0 ≥ 0 G(t) = gij (t) yiy(t) i (t Hawkes,Journal Journalofofthe theRoyal RoyalStatistical StatisticalSoc So A.A.Hawkes, Its dynamics is defined in terms of the linear integral equation Z ∞ y(t) = y0 + G(τ )s(t − τ ) dτ = y0 + (G ∗ s)(t). −∞ Hawkes, 1971 Stationary firing rates Assuming stationarity y(t) = y, one has Z ∞ y = y0 + G(τ )y dτ = y0 + Gy −∞ with G= Z ∞ G(τ ) dτ. −∞ If the matrix 1 − G is invertible we have y = [1 − G]−1 y0 . If, in addition, |λ| < 1 for all eigenvalues λ of G, the usual geometric series expansion suggests a decomposition into contributions of recurrent pathways of all orders y = [1 − G]−1 y0 = ∞ hX n=0 i Gn y0 = y0 + Gy0 + G2 y0 + . . . Stationary correlations Assuming joint stationarity, the pulse-coded interactions are conveniently quantified by the covariance functions cij (τ ) = Cov si (t + τ ), sj (t) = si (t)sj (t + τ ) − si (t) sj (t) . Using Fourier transforms Z ∞ ˆ f (ω) = f (t)e−iωt dt and −∞ fˆ(0) = Z ∞ f (t) dt −∞ Using Wiener-Hopf theory, Hawkes (1971) obtained for linearly interacting point processes −1 −1 ĉij (ω) = Ĉ(ω) = 1 − Ĝ(ω) Y 1 − Ĝ(ω)∗ where Y = diag(y). For the integrated covariances, one gets C = Ĉ(0) = 1−G −1 −1 Y 1 − GT . Expanding correlations Influence of direct and indirect connections Assuming again |λ| < 1 for all eigenvalues λ of G, the expansion If absolute value of eigenvalues of Gthe < 1,covariance Y =1 of [1 − G]−1 can be exploited to re-write (Y = 1) C = (1 − G )−1 (1 − G T )−1 = (1 + G + G 2 ...)(1 + G T + (G T )2 ...) ∞ hX ∞ ih X m i ∞ T� C = [1 − G]= 1 +[1G−+ G = G T ] T 2 G T 2 G + GG + G + (G ) + ... = G m (G T )n −1 T −1 n n=0 = 1 + G + GT + G 2 + G T =G + G 2 + G + T 2 GT2 m=0 m,n=0 T + GG T + . . . + GG + . . . +1 with matrix elements e.g. � � X TX [G 2 ]ij = Gik G(1,1) G G (2,0) kj 2 T [GG ]ij = gij ≡ G ij = gik gkj , gij ≡ GG ij = gikk gjkik , jk. . . k (Pernice, k Staude, Cardanobile, Rotter 2011, compare Rangan k PRE 2009) corresponding to shared input contributed by different types of multi-synaptic pathways (motifs). Pernice et al., 2011 Shared input motifs contributing to correlations Fluctuations of population activity Fluctuations of stationary population activity X S(t) = si (t) i can be expanded into contributions from all auto- and cross-covariances X Cov si (t + τ ), sj (t) . Cov S(t + τ ), S(t) = ij For a Hawkes process, we can exploit the power series expansion for the matrix of integrated covariances X X X (n,m) X X (n,m) X Cij = gij = gij = N 2 g (n,m) . ij ij nm nm ij nm 5 Negative feedback decorrelates activity g(m,n) g(m,n) B 0.010 0.005 0.000 0.005 0.08 0.04 0.00 0.04 theoretical mean random output mean fixed output mean p = 0.1 p = 0.25 1.0 Average correlations in (almost) regular networks do not depend on fine-scale structure. For dominant recurrent inhibition, motifs of uneven order m + n contribute negatively. Non-regular networks: cliques of excitatory hubs Patchy and/or cell-type specific connectivity dence on self-inhibition from linear theory, Fig. 3. Correlations in networks of LIFthe neurons An increase of Vθ causes weaker impulse responses and ĉij (ω = 0) cross-correlation FIG. 2. (color (a) Scatter plot of covariances between Re[ĉijonline) (ω = 310 Hz)] Im[ĉ 310 Hz)] ij (ω = C neuron pairs (red: kj (ω = 0), green: Im[Ckj ](ω = 0.31kHz), Pernice et al., Phys Rev E 2012 black: Re[Ckj ](ω = 0.31kHz)). Inset: Standard deviation Higher-order cumulants A D A D A D A D Time Multivariate Hawkes process B B C B B Hawkes, Oakes, J Appl Prob, 1974 Saichev, Sornette, Phys Rev E, 2014 Jovanović, Hertz, Rotter, Phys Rev E, 2015 C C C A B C D e.g. the classic text by Felsenstein [27]), the numbe terms grows very quickly with increasing n (see Figur - source [27]) and thus computing k i (t) quickly becom impractical. A combinatorial challenge n=2 u1 t1 n=3 t2 u1 u1 u2 t1 t2 n=4 t3 t1 t2 t3 u1 u1 u1 u2 u2 u2 u3 u3 t1 t2 t3 t4 t1 t2 u1 t3 t4 t3 t4 u1 u2 t1 t2 t3 t4 t1 t2 t1 t2 t3 t4 n Terms in nth order density 2 1 3 4 4 26 5 236 6 2, 752 7 39, 208 8 660, 302 9 12, 818, 912 10 282, 137, 824 11 6, 939, 897, 856 12 188, 666, 182, 784 13 5, 617, 349, 020, 544 14 181, 790, 703, 209, 728 15 6, 353, 726, 042, 486, 272 16 238, 513, 970, 965, 257, 728 17 9, 571, 020, 586, 419, 012, 608 18 408, 837, 905, 660, 444, 010, 496 19 18, 522, 305, 410, 364, 986, 906, 624 20 887, 094, 711, 304, 119, 347, 388, 416 FIG. 3.Jovanović, NumberHertz, of terms in Phys ki (t)Rev forE,a 2015 given n Rotter, Dynamic neuronal networks of the brain Correlations and population signals Disentangling multi-synaptic pathways Inferring connectivity from correlations Inferring directions from a symmetric matrix? Pernice & Rotter, JSTAT, 2013 Fundamental degeneracy of the problem Given the covariance matrix Ĉ(ω), can one solve the equation Ĉ(ω) = 1 − Ĝ(ω) −1 −1 Y 1 − Ĝ(ω)∗ for the connectivity matrix Ĝ(ω)? The Idea followed here is to determine “some” square root of the inverse covariance matrix Ĉ −1 (ω) = B̂(ω)∗ B̂(ω) and extract Ĝ(ω) from the relation (assuming Y is known) B̂(ω) = Y −1/2 1 − Ĝ(ω) . However, for any unitary matrix U satisfying U ∗ U = U U ∗ = 1 the matrix  = U B̂ provides an equivalent solution Â∗  = B̂ ∗ U ∗ U B̂ = B̂ ∗ B̂. Searching for sparse networks (a) 0 0.05 (b) 0 (c) 0.05 0 5 5 10 0.00 10 0.00 10 neuron 5 15 (d) 0 0 5 10 15 0.05 15 (e) 0.05 0 10 0.00 10 15 15 neuron 5 5 10 15 neuron 0.05 10 15 0 5 0 5 0.05 FN FP 0 5 10 15 neuron TN 0.00 15 (f) TP 0.05 0.05 0 5 10 15 0 10 20 30 40 0 10 20 30 40 neuron TP FN FP TN Method: stochastic minimization of the L1 -norm of the coupling matrix, using the Cholesky decomposition to initiate the search Inference of connectivity from noise-free Ĉ(0) 1.2 / [G(0)] 1.0 n=50 (b) n=300 gE =0.005 gE =0.01 gE =0.02 gE =0.05 0.9 0.8 0.6 0.4 0.2 0.00.0 0.1 0.2 0.3 p (c) 1.0 true positive rate (a) 0.8 0.7 0.6 0.5 0.4 0.3 0.0 0.1 0.2 0.3 p 0.20.0 p=0.28 p=0.22 p=0.1 p=0.04 0.2 0.4 0.6 0.8 false positive rate Inference of connectivity from estimated Ĉ(ω) n=100 (b) 1.0 0.6 0.4 0.2 0.9 0.8 20 ×103 s 5 ×103 s 2 ×103 s 20 ×103 s 5 ×103 s 2 ×103 s true positive rate true positive rate 0.8 (c) 1.0 n=300 0.6 0.4 area under ROC (a) 1.0 0.2 0.00.0 0.2 0.4 0.6 0.8 1.0 0.00.0 0.2 0.4 0.6 0.8 1.0 false positive rate false positive rate 0.8 0.7 0.6 0.5 0.2 n=100, all n=300, all n=100, =0 n=300, =0 tsim [ms] 2.0 1e4 tate xt , consisting of hand x-/y-position and velocity, how evolved to influence that activity by visual ccording to the following linear stochastic difference equation: stimulation. New applications: “networks of networks” State model: t+1 (1) activity = MUA F · xt + omulti-unit + wt LFP local field potentials where wt is a zero-mean Gaussian noise process with covariance matrixECoG Q and o aelectro-corticogram constant offset. In addition, we assumed that MREG MR-encephalography he observed neuronal activity depended linearly on the hand tate according to(“gradient-less the following linear equation: imaging”) NIRS Near-infrared spectroscopy Pistohl et al., 2008 10 mm 10 µm Hubel, 1995 The visual cortex in a monkey, stained by the Go tate xt , consisting of hand x-/y-position and velocity, how evolved to influence that activity by visual ccording to the following linear stochastic difference equation: stimulation. New applications: “networks of networks” State model: t+1 (1) = F · xt + o + w t where wt is a zero-mean Gaussian noise process with covariance matrix Q and o a constant offset. In addition, we assumed that he observed neuronal activity depended linearly on the hand tate according to the following linear equation: Pistohl et al., 2008 10 mm 10 µm Hubel, 1995 The visual cortex in a monkey, stained by the Go Continuous variable systems Consider a multi-component system, with state variables y(t), driven by fluctuating input x(t). Assume that the system is characterized by a coupling matrix of response kernels G(t) and satisfies the linear consistency equation y(t) = x(t) + (G ∗ y)(t). In the Fourier domain, after taking expectations, we have and −1 ŷ = 1 − Ĝ x̂ ∗ −1 ∗ −1 ŷ ŷ = 1 − Ĝ x̂x̂ 1 − Ĝ∗ . In the case of the Hawkes process, the source term x̂x̂∗ is replaced by Ŷ . In other words, the spikes of the neurons in the network generate their own “driving noise”. Application to ECoG data Ball et al., 2009; Derix et al., 2012 Efficient inference of connectivity Niederbühl, Diploma thesis, 2014 Different frequency channels in ECoG Niederbühl, Schiefer, work in progress Application to MREG data Lee, Zahneisen, Hugger, LeVan, Hennig, NeuroImage, 2013 Pernice, Niederbühl, LeVan, work in progress Application to MREG data Lee, Zahneisen, Hugger, LeVan, Hennig, NeuroImage, 2013 Pernice, Niederbühl, LeVan, work in progress Application to MREG data Lee, Zahneisen, Hugger, LeVan, Hennig, NeuroImage, 2013 Pernice, Niederbühl, LeVan, work in progress Conclusions I Linear Hawkes processes are useful models for networks of irregularly spiking neurons. Recurrent networks of LIF neurons can be matched to an equivalent Hawkes process. We expect that this also applies to real nerve cells in neocortical networks. I Dynamic properties of spiking networks of finite size with arbitrary topology can be inferred analytically, using matrix algebra, provided couplings are weak and spike trains are irregular. I Pairwise correlations in large networks have strong impact on the amplitude of population signals (LFP, ECoG, EEG, MREG, fMRI). The contribution of specific multi-synaptic pathways (motifs) to pairwise and higher-order correlations can be computed for any given network. I The micro-topology of sparse networks can be approximately recovered from the covariance matrix, employing compressed sensing methods. Directed links can be inferred from non-directed zero-lag covariances, but accounting for temporal information improves the inference. Stefano Alexander Volker Jonathan John Stojan Marcel Benjamin Jürgen Pierre Tonio Markus Sina Cardanobile Niederbühl Pernice Schiefer Hertz Jovanović Sauerbier Staude Hennig LeVan Ball Kern Bert Thanks!