Inferring Neural Connectivity and the Underlying Network Dynamics from Spike Train Recordings

yNeuroscience Laboratory, Department of Applied Mathematics, School of Optics, Universidad Complutense de Madrid;Avda. Arcos de Jalon s/n, 28037, Spain.zLaboratory of Nonlinear Systems, Swiss Federal Institute of Technology Lausanne;EPFL-IC-LANOS, CH-1015 Lausanne, Switzerland.Email: vmakarov@opt.ucm. es, oscar.defeo@epﬂ. ch, ﬁvos. panetsos@opt.ucm. esAbstract—A novel method for the identiﬁcation and mod-eling of neural networks using experimental spike trains isdiscussed. The method assumes a reference model of inter-connected deterministic integrate-and-ﬁre neurons and ﬁtthe parameters of the model to the observed experimentalspike trains. The identiﬁcation provides the properties ofthe individual synapses and neurons, hence extracting thefunctional connectivity between neurons. The method isshown to be e ective when applied on simulated data.1. IntroductionThe qualitative and quantitative analysis of the spiking ac-tivity of individual neurons is a very valuable tool for thestudy of the dynamics and functional architecture of theneural networks in the Central Nervous System [1]. In par-ticular, deducing the functional connectivity of neural net-works from experimental data, usually restricted to spiketrains, is of crucial importance in neuroscience: for thecorrect interpretation of the electrophysiological activity ofthe involved neurons and networks; and, more important,for correctly relating the electrophysiological activity to thefunctional tasks accomplished by the network. Here, theterm functional stands for any observable, direct or indi-rect, interaction between neurons which alters their spiketimings.The measured activity of a neuron is not the result ofits solely intrinsic properties, but stems from the directand indirect inﬂuences of the other neurons of the net-work, leading to network behaviors far beyond the sim-ple combinations of those of the isolated neurons. On theother hand, the measured time instants of spike occurrences(point events) do not allow any direct insights about thesubthreshold and/or intrinsic membrane dynamics of theneurons. Nonetheless, spike trains can be used to identifythe functional characteristics and e ective architecture ofthe neural network they originated from, e.g [2, 3]. The most common and standard methods for identifyingthe synaptic connections between neurons assume a sto-chastic nature of the spike trains, and the functional bondbetween two neurons is extracted from the statistical infer-ence of the discharges times, usually deducing it from theshapes of cross-correlograms [2, 3]. Though widely under-stood, this tool provides a very limited knowledge aboutthe functional properties of the neural networks, and itcannot distinguish direct from indirect connections amongneurons. Recently, more sophisticated statistical methods[4, 5, 6] have overtaken this problem. However, these meth-ods still ﬁt into a stochastic framework and lack a com-pact description of the estimated interactions. Furthermore,since they assume a stochastic nature of the spike trains,they do not provide considerations about the dynamics ofthe involved neurons, nor about the nature of the intrinsicprocesses that are responsible for such behavior.In contrast to a purely statistic approach, a deterministicone can be considered, with the main advantage of pro-viding a mathematical model for inferring single neuronor neural network properties indirectly. In this direction,methods for extracting a dynamical system out of the inter-spike intervals have been recently proposed [7, 8], howeverthese methods assume neurons to be isolated; hence, theydo not provide insights about the neural network structureand its relationships with the observed dynamics.Here a new model based method for the identiﬁcationand modeling of whole neural networks from experimentalspike trains is proposed. A description of the method isgiven in Sec. 2, whilst in Sec. 3 numerical tests of it arepresented and then discussed in Sec. 4.2. Identiﬁcation MethodThe identiﬁcation method, despite being quite mathemati-cally convoluted, is rather transparent in its principle. Thereference model adopted in the identiﬁcation process is anetwork of interconnected integrate-and-ﬁre neurons. Allthe parameter values necessary to univocally deﬁne it, i.e.the connectivity matrix of the network, the synaptic timescales, and the intrinsic parameters of the neurons, are de-rived from the recorded spike trains by an optimizationprocedure which minimizes the di erence between the pre-dicted and measured timings of spike episodes.Precisely, given N spike series from as many neurons,the reference model is a network composed of N intercon-nected single-compartment leaky integrate-and-ﬁre mod-els. The connections between neurons are represented by aN N matrix W whose elements w