This volume contains extended abstracts outlining selected talks and other selected presentations given by participants throughout theWe use the piecewise linear McKean model to present a proof-of-concept to address the estimation of synaptic conductances when a neuron is spiking. Using standard techniques of non-smooth dynamical systems, we obtain an approximation of the period in terms of the parameters of the system which allows to estimate the steady synaptic conductance of the spiking neuron. The method gives also fairly good estimations when the synaptic conductances vary slowly in timePeer ReviewedPostprint (published version

Guillamon Grabolosa, Antoni

Prohens Sastre, Rafel

Teruel Aguilar, Antonio E.

Vich Llompart, Catalina

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Nonlinear Estimation of SynapticConductances via Piecewise Linear SystemsAntoni Guillamon, Rafel Prohens, Antonio E. Teruel, and Catalina VichAbstract We use the piecewise linear McKean model to present a proof-of-conceptto address the estimation of synaptic conductances when a neuron is spiking. Usingstandard techniques of non-smooth dynamical systems, we obtain an approximationof the period in terms of the parameters of the system which allows to estimate thesteady synaptic conductance of the spiking neuron. The method gives also fairlygood estimations when the synaptic conductances vary slowly in time.1 IntroductionThe problem addressed in this manuscript is framed into the challenge of unveilingthe functional connectivity in the brain; that is, to obtain information about strengthand timing of the input currents (mediated by synapses) that a single neuron (thepost-synaptic cell) receives from others (the pre-synaptic ones). In particular, weaim at disentangling the arrangement of excitation versus inhibition impinging onthe cell.Wewill focus on the estimation of the conductances of these currents, a quan-tity which is not directly measurable but needs to be extracted from experimentallyA. Guillamon (B)Departament de Matemàtiques-EPSEB, Universitat Politècnica de Catalunya,Barcelona, Spaine-mail: antoni.guillamon@upc.eduR. Prohens · A.E. Teruel · C. VichDepartament de Matemàtiques i Informàtica, Universitat de les Illes Balears,Palma, Spaine-mail: rafel.prohens@uib.catA.E. Teruele-mail: antonioe.teruel@uib.esC. Viche-mail: catalina.vich@uib.es© Springer International Publishing AG 2017A. Colombo et al. (eds.), Extended Abstracts Spring 2016,Trends in Mathematics 8, DOI 10.1007/978-3-319-55642-0_168990 A. Guillamon et al.accessible data like themembrane potential of the neuron. For a review of the relevantconcepts in neuroscience, we refer the reader to the book Ermentrout–Terman [4],and the introduction in Vich–Guillamon [8] for a quick survey on the problem ofestimation of conductances. A major hindrance in this problem is the estimationwhen ionic currents are active, specially in spiking regimes, where ionic conduc-tances are much higher than synaptic ones; see Guillamon–McLaughlin–Rinzel [6].Nomethods are yet known to estimate synaptic conductances in this paradigm. Fromamathematical point of view, this is indeed an inverse problem consisting of estimat-ing some parameters from time courses of one variable of the associated dynamicalsystem. In this work, we aim at giving a theoretical proof-of-concept by consid-ering a simplified model of neuronal activity, namely a non-smooth caricature ofthe FitzHugh–Nagumo model (see Coombes–Thul–Wedgwood [3]), which allows avery sharp approximation of the nonlinear f − I curve thanks to special propertiesof piecewise linear systems.Moreover, the same idea could be applied to smooth sys-tems with approximate expressions of the corresponding f − I curves. The modelwe consider is a version of the McKean model as described in Coombes [2], wherethe synaptic current, Isyn(v), has been considered apart from the rest of externalcurrents, I . That is, we consider the system{C v˙ = f (v)− w − w0 + I − Isyn(v),w˙ = v − γw − v0, (1)where Isyn(v) = gsyn(v − vsyn) and f (v) is defined in a continuous butnon-differentiable way according to three zones of the phase space, left, middleand right: f (v) = fL(v) := −v if v < a/2, f (v) = fM(v) := v − a if a/2 ≤ v ≤(1+ a)/2, and f (v) = fR(v) := 1− v if v > (1+ a)/2. For the significance of theparameters involved in the model, see Coombes [2]. Here, we focus on the mostrelevant parameters for this work. We consider to have a single source of synapticconductances, which is modeled by means of vsyn , the synaptic reversal potential,and gsyn > 0, the synaptic conductance. The ionic currents are modeled through theexpression f (v)− w − w0 in the first equation of (1), designed to mimic the quali-tative behaviour of the N-shaped v-nullcline of more biophysically realistic models.Since C , the parameter related to cell membrane capacitance, is assumed to be smalland bounded, 0 < C ≪ 0.1, system (1) is then a slow-fast dynamical system, wherev is the fast variable and w is the slow one.In Abbott [1] and Tonnelier–Gerstner [7], the authors show the existence of twovalues I1 < I2 for the applied current forwhich system (1)with Isyn(v) = 0 exhibits aperiodic orbit, which is unique, if and only if I1 < I < I2. In Coombes [2], approx-imations of the period T were obtained, mainly based on the slow-fast nature ofthe system.More recently, in Fernández-García–Desroches–Krupa–Clément [5], theauthors provide an approximate expression for the period T by taking advantage ofthe slow invariant manifolds for C ≪ 1. We show that the previous scenario per-sists for Isyn(v) ̸= 0 and that the interval where periodic orbits exist depends also ongsyn as well as the period, T = T (C, I, gsyn). We present a new approximation ofNonlinear Estimation of Synaptic Conductances … 91T , Tˆ , which includes this dependence on gsyn and, moreover, improves the existingapproximations. The key points are two refinements: the consideration of the flighttime in the central region, and a special projection on the slow manifold.In the range of applied currents in which the neuron is spiking, we have foundnumerical evidence that the period of this oscillation has a nonlinear but monotonicdependence on gsyn . Hence, as a consequence of this monotonicity, by knowing Tˆand the applied current I (i.e., knowing the so-called f − I curve), we can computegsyn by solving numerically a non-linear equation with a unique solution, and thuswe are able to estimate the steady synaptic conductance of the neuron, which is thegoal of this work.2 Main ResultIt is easy to check that the system (1) with f (v) = fL(v) for all v has a unique fixedpoint, pL ; similarly, considering f (v) = fM(v) and f (v) = fR(v) respectively, wedefine pM and pR . When a limit cycle exists, both pL a pR , which have the sameJacobian matrix, lie on the middle zone and so, they are only virtual equilibria;however, their eigenvalues have influence in the dynamics. We denote by λq/s,L/Mthe eigenvalues of the equilibrium points pL and pM of system (1), where q and sstand for “fast” (eigenvalue with the biggest modulus) and “slow”, respectively. Toensure the existence of a periodic orbit,we need to consider the next set of hypotheses:gsyn > 1− γ−1, |gsyn + Cγ| < 1, 0 < C ≤ C∗, and I1 < I < I2, (H)whereI1 =(a2− vsyn)gsyn + (γ + 1)a − 2v0 + 2γw02γ , I2 = I1 +12gsyn + 1− γ2 γ .(2)With these definitions, we can state the following result.Theorem 1 Given system (1) under hypothesis (H), for a sufficiently small C > 0,the period of the unique periodic orbit of the systemcanbewritten as T = Tˆ + O(C),withTˆ =2∑j=11λs,Lln(∣∣∣∣∣ γ(I − I j )Blγ(I − I j )Bl + (−1) j Kl∣∣∣∣∣)+ 1λq,Mln(∣∣∣∣ γ(I − I j )Bm + Kmγ(I − I j )Bm + Km, j∣∣∣∣) ,where92 A. Guillamon et al.Bm = (γ + λq,M )((Gsyn + 1)(λs,L − λs,M )− γ(λs,L + λs,M )− 2λs,Lλs,M),Bl = λq,L − λs,L ,Kl = 12 (γ + λq,L )(Gsyn + γ + 2λs,L + 1),Km = 12 (γ + λs,M )(Gsyn − γ + 1)((Gsyn + 1)(λs,L − λq,M )− γ(λs,L + λq,M )− 2λs,Lλq,M),Km,1 = 12 (γ + λq,M )(Gsyn − γ + 1)((Gsyn + 1)(λs,L − λs,M )− γ(λs,L + λs,M )− 2λs,Lλs,M),Km,2 = 12 (λq,M − λs,M )(γ + λq,M )(Gsyn − γ + 1)(Gsyn + γ + 2λs,L + 1),Gsyn = gsynγ.Sketch of the proof The period is obtained through straightforward computationswhich are based on the local linearity of the vector field and the existence of twopoints, on the switching line, belonging to the periodic orbit. These two points areapproximated as the intersection of the slow manifold with the switching line, sincethe periodic orbit is assumed to pass exponentially close of the slow manifold. !3 Application to the Estimation of ConductancesAll our numerical tests show evidences that Tˆ is monotonous with respect to gsyn inthe spiking region I ∈ (I1, I2); see (2). We have not been able to prove this propertyyet, but it works for practical purposes and allows us to compute good estimationsof slowly-varying synaptic conductances in spiking regimes. It is quite obvious thatforcing the system (1) with a steady conductance, the system will exhibit a periodicbehaviour of period T and, therefore, solving the equation Tˆ (C, I, gsyn) = T , wewillrecover the input synaptic conductance with high accuracy. The challenge, however,is to know whether the conductances can be well estimated when they fluctuate. Totest it, we have forced the system (1) with prescribed conductance traces, gsyn(t),and then proceed as follows:(i) We extract the interspike intervals (distances from maxima) from the resultingvoltage trace, thus obtaining a sequence of fluctuating periods {Ti }Ni=1, whereN + 1 is the number of spikes in the simulation.(ii) For each i = 1, . . . , N , we have solved (numerically) the equation Tˆ (C, I, gsyn)= Ti and choose the positive solution gsyn,i satisfying I1(gsyn,i ) < I < I2(gsyn,i )to ensure that corresponds to a periodic orbit.In Fig. 1, we show estimations obtained from three different prescribed inputs: asinusoidal drive, an oscillatory drive with two frequencies, and a stereotypical synap-tic input to a single cell in visual cortex (units scaled to match (1)). We observe fairlygood estimations of synaptic conductances (see panels (a), (b), (d) and (e) in Fig. 1).Comparing panels (a) and (b), we observe that, including a higher frequency in theprescribed input, the estimations impoverish. In general, frequencies higher thanthe sampling frequency are difficult to estimate. However, the reconstructed voltageNonlinear Estimation of Synaptic Conductances … 93(a) (b) (c)(d) (e)Fig. 1 Estimation of synaptic conductances for different inputs. (a, d, e) estimations of the timecourse of gsyn when the neuron’s synaptic input is a sinusoidal drive, a two-frequency oscilla-tory drive and a conductance trace taken from a realistic model, respectively; the time coursesof prescribed conductances appear in black, open circles indicate the estimates obtained with ouralgorithm and the spline interpolations of these estimates are plotted in red; (b) estimated valuesof gsyn against the prescribed values for the sinusoidal drive; (c) voltage trace using the prescribedconductances for the sinusoidal drive (black) compared to the voltage trace reconstructed using theestimated conductances (red)using the estimated conductance (see panel (c)) exhibits a good agreement in bothcases (only sinusoidal drive is shown). Notice that when the prescribed conductancesare taken from a realistic model of V1 (see panel (e)), the estimation captures theoverall conductance profile but does not match at a smaller scale. In realistic inputs,the conductances’ time-scale is variable, being very short in some moments. Thus,we can only get a good estimation on average. A further improvement of our esti-mation method will consist of introducing more Poincaré sections to estimate flighttimes in shorter intervals (i.e., refining the sampling).Acknowledgements This work is partially supported by the Spanish Ministry of Economy andCompetitiveness through project MTM2015-71509-C2-2-R (AG), by the MCYT/FEDER grantnumber MTM2014-54275-P (RP, AT and CV) and by the Government of Catalonia under grant2014-SGR–504 (AG).References1. L.F. Abbott, A network of oscillators. J. Phys. A: Math General 23(16), 3835 (1990)2. S. Coombes, Neuronal networks with gap junctions: a study of piecewise linear planar neuronmodels. SIAM J. Appl. Dyn. Syst. 7(3), 1101–1129 (2008)3. S. Coombes, R. Thul, K.C.A. Wedgwood, Nonsmooth dynamics in spiking neuron models.Phys. D 241(22), 2042–2057 (2012)94 A. Guillamon et al.4. G.B. Ermentrout, D.H. Terman, Mathematical Foundations of Neuroscience (Springer, NewYork, 2010)5. S. Fernández-García, M. Desroches, M. Krupa, F. Clément, A multiple time scale couplingof piecewise linear oscillators. Application to a neuroendocrine system. SIAM J. Appl. Dyn.Syst. 14, 643–673 (2015)6. A. Guillamon, D.W. McLaughlin, J. Rinzel, Estimation of synaptic conductances. J. Physiol.-Paris 100(1–3), 31–42 (2006)7. A. Tonnelier, W. Gerstner, Piecewise linear differential equations and integrate-and-fire neu-rons: insights from two-dimensional membrane models. Phys. Rev. E 67, 021908 (2003)8. C. Vich, A. Guillamon, Dissecting estimation of conductances in subthreshold regimes. J.Comput. Neurosci. 39(3), 271–287 (2015)

Nonlinear estimation of synaptic conductances via piecewise linear systems

This volume contains extended abstracts outlining selected talks and other selected presentations given by participants throughout theWe use the piecewise linear McKean model to present a proof-of-concept to address the estimation of synaptic conductances when a neuron is spiking. Using standard techniques of non-smooth dynamical systems, we obtain an approximation of the period in terms of the parameters of the system which allows to estimate the steady synaptic conductance of the spiking neuron. The method gives also fairly good estimations when the synaptic conductances vary slowly in timePeer Reviewe

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Nonlinear estimation of synaptic conductances via piecewise linear systems

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