A Reduced Model for Dendritic Trees with Active Membrane

Transcription

1 A Reduced Model for Dendritic Trees with Active Membrane M. Ohme and A. Schierwagen Institute for Computer Science, University of Leipzig, D Leipzig Abstract. A non-uniform equivalent cable model of membrane voltage changes in branching active dendritic trees has been developed. A general branching condition is formulated, extending Rall's 3/2 power rule for passive trees. 1 Introduction The basic processing units in current models of artificial neural networks (ANN) are mostly poor caricatures of real neurons, although various authors have often stressed the importance of dendritic signal processing for neuron function (see [2] for a short overview). One reason may be the lack of tools to analyze complex branched dendritic trees - only simulations are possible as yet. Especially for network studies reduced models are desirable. A powerful tool for analyzing passive dendrites is the reduction to an equivalent cable [4] by means of the linear cable theory. Provided certain symmetry conditions are fulfilled, such a tree can be treated as a single cable, and for special cases of the underlying geometry analytical solutions can be derived [5]. However, for several important classes of vertebrate neurons, active ion channels have been positively demonstrated in the dendrites or at least strongly suggested [2; 6]. Also, the reduction to an equivalent cylinder could be helpful for describing signal flow in axonal arbors. We will show here that the reduction to an equivalent cable is also applicable for trees with active membrane. Starting with the general cable model for dendritic segments we derive branching conditions for those trees which can be collapsed into only one single equivalent cable. These symmetry conditions turn out to be similar to the passive case. 2 Mathematical model of a dendritic tree General cable model for dendritic segments. The usual mathematical description of the potential distribution in dendritic trees proceeds from the application of one-dimensional cable theory to the tree segments. Representing the cable by a RC ladder network, the cable equation for the transmembrane potential V(x, t) and the axial current ja(x, t) in a single segment is as follows (x represents distance in axial direction): 1 OV Oj~ J~ - r~(x) Ox' -- ox = jm = Je + ji (1) where j,~(x, t) denotes the membrane current consisting of a capacitive component jc and a resistive one ji- We assume that the current ji created by the ionic channels in the membrane can be split into a product of a "resting" conductance which depends on the

2 692 membrane surface at x and a nonlinear potential function fo(x, V, ltl,..., Un) reflecting the threshold behavior of the voltage dependent channels as a specific membrane property (i.e. per unit membrane surface). The latter may be time depending so additional variables uk defined by further first order differential equations have to be included: (. OV jc = c x)-~ ji = g(x)fo(x,v, Itl,...,Itn) Ouk --fk(x,v, ul,...,un) Ot forl<k<n. Combining equations (1) and (2a) we obtain for every segment 0 1 OV) OV Ox r~-;- "(x) Ox = jm = jc + ji = c(x)--x7'o/: -[- g(x)f~ Y' Ul' " " ' un) (2a) (25) (3) Applying the variable transform t T = - T with X = fo = ~-=-_ ds with T= - c(x) g(x) 1 jro(x)g(x)' and (4a) (4b) equation (3) can be rewritten in dimension-less coordinates 02V OV OV o= +Qax at fo(x,v, OUk -Tfk(X,V, ul,...,un) for l<k<n (5) OT 1 0 [ g(x) with Q(X) = ~ 0---X in \r~(x) ] / g(x) ov The axial current j~ then is given by ja(x,t) = -V r~(z) 0X (6) Representation of the neuronal tree structure. To represent a reconstructed dendritic tree, we use the graph theoretical concept of a directed, labeled tree. All branching points of the dendrite are then represented by the nodes of the tree with the soma as root point. Some (virtual) branching points have to be included for additional localized input into the dendrite. The edges of the tree correspond to the single dendritic segments. Any edge 7r is labeled with the quantities necessary to set up the cable equation for the corresponding cable segment, i.e. length l~, number n~ of auxiliary variables Uk, functions ra~, g~, c~, f~ and f~rk. Note that the origin of the space variables x and X is the root node. External current input can enter the tree only by the nodes. So every node K has to be labeled with a current function IK which depends on the potential (and its time derivatives) at this site, and explicitly from the time. Two simple

3 693 examples are (1) synaptical input modeled by a change of synaptic conductance and (2) a lumped soma modeled by a RC circuit: IK(t,V) = [0 ift<a I~;(V, Vt) = 1-~--V [ ~-~ (t - A) exp(p~) (~ - V) ' RK +CgVt From Kirchhoff's first law (saying that the sum over all currents flowing into a node K vanishes), we yield I~: + E,c~(K) ja~(xk, t) -- japs(k)(x~:, t) = O, (7) where CS(K) denotes the set of all child segments ~ with K as starting node and 7)$(K) stands for the (unique) parent segment w with K as end node. X/~ is the electrotonical distance of the the node K from the soma. To find the potential and current distribution over a given neuronal tree to some external input functions IK one has to solve equation (5) for all segments under the above boundary conditions (7). 3 Reduced model for an active dendrite Obviously, the above problem is in general hard to solve. In most cases the only way to get some insight in possible reactions of the system lies in numerical simulations which often remain unsatisfactory because of too many free parameters. Thus any reduction of the original problem to one with less free parameters would be of great help. We will show that the reduction approach for passive dendrites [4] can be generalized to active trees, too. This can be formulated as Theorem (equivalent cable). There is a cable which is equivalent to the whole t~e in the sense that every solution (potential and current distribution) for this equivalent cable leads to a solution for the tree provided some conditions given below are fulfilled. This means here that the voltage change in time at any point X, in the tree is equal to the potential change of the equivalent cable at point X with the same electrotonie space coordinate. The sum of axial current flowing through all segments with equal electrotonical distance X from the root is equal to the axial current of the equivalent cable at X. The analog is true for the membrane current between two space coordinates X1 and X2. The equivalent cable will be described by the same cable equations (5) with parameters leq, raeq, g.~, coq, ne~, re, and fr Then the conditions for reduction of a neuronal tree are the following: time constant: T(X) has to be constant and independent from the segments and equal to ~-o~. equal length: All terminal nodes T have the same electrotonical distance L from the root S: XT = fo r dx -- L = L~. equal cable model: The functions fri are the same for any segment: f~i(~/~,y, Ul,...,Un) : f~qi(x,y, Ul,...,Un). similar cable geometry (generalized ~- branching rule): The same has to be true for the geometrical parameter Q~(X) = Qo~(X), i.e. there are constants

4 694 C~ for every segment 77 with = (8) At the nodes of the tree an additional condition must be satisfied: the constant C, belonging to a parent segment 7r (with CT,s(s) := 1 for the nonexistent parent segment of the root) has to be equal to the sum of the C~ which belong to all child segments w of 7r: 1 C, =Ewecs(~)C~ (9) similar boundary conditions: External input into the tree has to be symmetrically divided among all places with the same electrotonical distance from the root. That is, for two nodes K and L with the same electrotonical distance XK = XL away from the root S the attached current functions have to fulfill: IK/C'ps(K) = [L/CT)S(L). (10) The input current I]r into the equivalent cable at the same electrotonical distance X = XK is defined by the sum of all inputs at equal electrotonic distance X: I~" := ExL=x IL. (11) Proof of theorem: First we state that any general potential solution of the equivalent cable yields a general solution for all segments ~r = (K, L) limited to the corresponding space interval [XK,XL]: VT >_ 0: V~(X,T) = V(X,T). This follows from the assumption that the relevant parameters in the normalized cable equations (5) are equal for all tree segments. To verify that a potential distribution over the tree which yield s from a special solution of the equivalent cable is also a special solution of the tree one has to compute the axial currents in all segments and to prove that the boundary conditions (7) are satisfied. It follows from the generalized ~ branching rule that for any electrotonical distance X the sum over all C~ is equal to 1 [3]. With this one gets for the input current relation at X with ~v = PS(K): IK = C~Ik", (12) resulting from the definition (11) of I:~ " and the condition (10). The axial current in segment ~ (see (6)) is computed by: ja~(x,t) = / geq(x) OV (8) C, / g(x) OV -Vrao~(X) OX - Vr-~) ox - C,j~oq(X,T) (13) Inserting (12) and (13), the boundary condition (7) yields (Tr = 7)3(/()): 1 In the case of a tree with cylindrical segments, i.e. g~ (X)/ra~ (X) o( d 3, and constant specific cell parameters Ri, Rm and Cm one gets Rall's ~ branching rule[4].

5 695 ol 21111nl :: ;i... ~V;~ ~'~:;.... ct;:~;:2-~... :... f 84 ii :... i 11: i _9 ~12t i I~ i Di~meler p~file i ~ i Digmeter profile, : 211 i 0.7~ ~... i i.... ] :i i... ii: i... ] i~ii... i i 1 II II ~ 0.7 ~i... :iill.: ii ~ ii i [!!! i : :!! i Fig. 1. Comparison of active excitation propagation in a branched neuronal structure {left) and in the corresponding equivalent cable (right). For each case it is shown a) the tree as a dendrogram (left half) in the original (top) and in the normalized space (bottom); b) the profile of the diameter (bottom right) and c) the results of the simulation (top right). ZK-'[-~-~ Jaw (XK, T)--Jalr (XK, T) -= CTrIf( q-[-~'~ Cwja~q (X+O, T)- CTrja~q (X-O, T) ~CC, S( K) wecs( K) (9 ) CTr (Ifi q + ja.q(x+ O,T) - CTrjaeq(X- O,T)) = 0 where =t=0 denotes the left and right limit respectively. So the last term in brackets is the first Kirchhoff's law for the equivalent cable at input site X which vanishes according to our premise. This completes the proof. 4 Example To illustrate the application of this approach we studied the signal flow in a neuronal structure and its equivalent cable by compartmental simulation (Fig. 1). All segments are described by the FitzHugh-Nagumo system [1], which has one (n = 1) auxiliary variable u and a cubic current-voltage relation f in (3). The electrical parameters r~, g and c result from the standard equations for a nonuniform cable of circular cross-section, here with fixed, slowly increasing (near the soma) or decreasing (at the terminals) exponential diameter function, i.e. Q(X) in (5) is constant in all cases. Essential for the existence of an equivalent cable are the equal electrotonical distance of the terminal nodes, the synaptical inputs (filled circles in Fig. 1) and the changes of the equation types (here only the segment geometry represented by Q(X) - changes from segment to segment) from the soma. The realization of this requirements can easily be seen by the representation of the tree in the normalized space (Fig. 1, bottom left). The cell receives synaptical input (filled circles in the dendrogram) near the dendritic tips (into segments 111, 112, 121, 2111, 2112 and 2121) with

6 696 strength proportional to the diameter at the corresponding place to the power (see equations (8) and (I0)). The "recording sites" are the soma and some intermediate points (open circles in the dendrogram). In the right top parts of Fig. 1 the resulting signals are shown. The solid lines stand for the potential function at the recording sites (A corresponds to the input sites, D to the soma, B and C combine all the intermediate recording sites). The differences of the potential functions at different recording sites with equal electrotonical distances to the soma are to small to see them in the plot. The same is true for differences between the full tree (left) and the equivalent cable model (right). If we restrict the input into the cell to only one injection side (synapse on segment iii), we get an excitation propagation to the soma (dashed lines A, B, C and D), similar to the first case, but at every branching point an impulse is send in the hitherto non-excited branches. This leads also to a small time delay for the somatic impulse. In Fig. 1 the potential is drawn only for some recording sites: A corresponds again to the injection site, B to the recording site in segment ii, C corresponding in i, D at the soma, E in 2, F in 211 and 212 as well as G in 2111, 2112 and Both records F at 211 and 212 and G at 2111, 2112 and 2121) are not distinguishable in the plot because the subtree of segment 2 can be modeled again by an equivalent cable. 5 Conclusions The aim of this study was to contribute to the problem of building reduced models. In the literature, Rall's equivalent cable model has been also employed for active trees, however, without strict justification. As an example we mention Mel's clusteron model which consists of a soma and a single dendritic segment. There has been shown that this model with Hebb-type learning rules is able to effectively perform memory tasks [2]. To provide theoretical justification for such reduced models, we developed an "equivalent cable" model for active dendrites, comprising Rall's model for passive trees as a special case. The reduction process has been demonstrated by a simulation of an arbitrarily constructed neuronal tree the segments of which were modeled by the FitzHugh-Nagumo system. In this way, the analysis and simulation of potential spread in rather realistically modeled neurons can be distinctly simplified. It could therefore be helpful in closing the gap between ANN models and biological neural structures. Acknowledgements Supported by Deutsche Forschungsgemeinschaft, Grant No. SCHI 333/4-1. References 1. R. FitzHugh. Mathematical models of excitation and propagation in nerve. In H. P. Schwan, editor, Biological Engineering, pages 1-85, McGraw-Hilh New York, Bartlett W. Mel. Information processing in dendritic trees. Neural Computation, 6, Michael Ohme. Modellierung der neuronalen Signalverarbeitung mittels kontinuierlicher Kabelmodelle. Dissertation, University of Leipzig, Department of Computer Science, W. Rall. Theory of physiological properties of dendrites. Ann. N. Y. Acad. Sci., 96, A. K. Schierwagen. A non-uniform equivalent cable model of membrane voltage changes in a passive dentritic tree. J. ~heor. Biol., 141: , I. Segev. Dendritic processing. In Michael A. Arbib, editor, The handbook of brain theory and neural networks. MIT Press, 1995.