Neural Bistability and Amplification Mediated by NMDA Receptors: Analysis of Stationary Equations

Transcription

1 Neural Bistability and Aplification Mediated by NMDA Receptors: Analysis of Stationary Equations Patrick A. Shoeaker a a Tanner Research, Inc., 825 South Myrtle Ave., Monrovia, CA 91941, USA Voice pat.shoeaker@tanner.co Abstract The acroscopic current/voltage relationship of NMDA receptor ion channels is nononotonic under physiological conditions, which can give rise to bistable and aplifying/facilitatory behavior in neurons and neural structures, supporting significant coputational priitives. Conditions under which bistable regies of operation prevail, and also general aplifying properties associated with active NMDA receptors, are exained in a single copartent enclosed by a cell ebrane, and subsequently in cablelike dendrites under varying boundary conditions. Methodology consists of nuerical and atheatical analyses of stationary versions of equations governing the electrical behavior of these systes. Bistability ediated by NMDA receptors requires interaction with other conductances in the ebrane or cytoplas, with particular iportance attached to ebrane potassiu conductance, especially that of inwardrectifying potassiu channels. A corollary conclusion is that coactivation of GABA B synaptic receptors or SK channels is a coputationally powerful and soeties necessary adjunct condition for NMDA receptor-ediated bistability. Neural ultistability due to dendritic bistability is considered, including in the case of closely coupled dendrites. The characteristics of coactivation-dependent facilitation, and aplifying states in which NMDA receptor activation boosts the efficacy of other classes of synapses, are also described. Coactive inward-rectifying potassiu channels are found to significantly affect the characteristics of such aplification. Keywords: NMDA receptor; bistability; neuron; synapse; dynaical systes; GABA B ; SK channels This is the web version of an article appearing in Neurocoputing: P.A. Shoeaker, Neural Bistability and Aplification Mediated by NMDA Receptors: Analysis of Stationary Equations, Neurocoputing, doi: /j.neuco , The original publication is available at:

2 Introduction This paper concerns electrical effects associated with the broadly distributed class of synaptic receptors (typically activated in vivo by the neurotransitter glutaate) for which N-ethyl-D-aspartate (NMDA) is an agonist. The acroscopic current-voltage relationship of the ion channels associated with NMDA receptors (NMDARs) is significantly nonlinear under physiological conditions [1,2], displaying a negative slope conductance regie due to (kinetically fast) agnesiu blockade of the channels. This feature renders it capable of supporting bistable and aplifying behavior in conjunction with other ebrane conductances and/or intracellular resistance. Bistability the capacity of a syste to exist in either of two stable states when at equilibriu is a phenoenon of interest in neuroscience because it provides a potential echanis for short-ter eory (NMDARs have in fact been iplicated in working eory in neuropsychology [3-5]), and ore generally for a class of coputations that iplicitly require eory, such as teporal integration [6-8]. In physiological studies, evidence consistent with bistable behavior at the whole-neuron level has been observed in vertebrate neurons including Purkinje cells [9,10], pyraidal neurons [11-13], itral cells in the olfactory bulb [14], entorhinal cortical cells [15], and various otoneurons [16-18]. Such behavior can arise as a network property in interconnected sets of neurons, due to positive feedback in cobination with the saturating electrical characteristics of neurons (see, e.g., Li et al. [19]). However, it can also occur at a ore priitive level, in individual neurons and their processes. The echanis at this level is a nononotonic dependence of current flowing through the cell ebrane on the potential difference across the ebrane. There are several physiological echaniss by which such a characteristic can arise, including the nonlinear acroscopic conductance of NMDARs, as well as the effects of slowly- or non-inactivating sodiu [20,21] and calciu [22-24] currents. Of these, the NMDAR-based echanis is particularly significant due to its direct control by synaptic inputs to a neuron. In a bistable neuron or neural process, a regie exists in at least part of the structure in which the relationship between an injected current and local ebrane potential is N- shaped and has three zeros of the current. The two zeros that occur with positive slope (those at the sallest and largest voltages) correspond to stable equilibria, and the iddle zero to an unstable equilibriu. If the ebrane current-voltage (I-V) relation itself has this characteristic, then when it encloses an isopotential electrical copartent, the syste will be bistable with equilibriu potentials at the values corresponding to the stable zeros of the ebrane current. I use the ter bistable ebrane I-V characteristic as an abbreviated reference for a ebrane current-voltage relationship of this sort, as distinct fro, say, a nononotonic I-V relationship that has only a single zero. The latter case is still of interest with regard to bistability in non-isopotential neural odels. In addition to bistability, NMDARs can also ediate aplifying effects in nerve cells and neural processes; such effects have been noted in various cortical neurons [25-33], and iplicated in the linearization of EPSP suation in CA1 pyraidal cells [34]. In qualitative ters, aplification associated with NMDARs is due to the fact that the ore depolarized a neural process becoes, the greater the net depolarizing current contributed

3 by active NMDAR channels, while the local ebrane potential reains within the negative conductance regie. This can increase the effectiveness of other classes of synaptic input; it can also result in pronounced superlinear effects in cobination with voltage-gated ion channels [35]. In addition, because a fraction of this current is carried by calciu ions [36,37], an additional possible aplifying echanis is the activation of calciu-dependent sodiu channels [38,39] by increase of intracellular calciu. NMDAR-ediated aplification provides a local echanis for nonlinear facilitation of perhaps equal possible iportance to bistability. Various workers have coented on the potential coputational significance of aplification associated with NMDARs [30,32,40,41], and Mel and colleagues have developed and studied coputational algoriths based on it [32,42-44]. Schiller and Schiller [30] defined three putative regies of the electrical effects associated with NMDARs: boosting, bistable, and self-triggering. The general conditions under which such regies prevail, however, have apparently not been exained in detail. Lazarewicz et al. [41] considered bistability induced by NMDARs in cobination with other ebrane conductances, focusing on a particular structure (an electrically isolated dendritic segent with local synaptic activation along a fraction of its length), for specific cobinations of geoetric and electrical paraeters. Korogod & Kulagina [45] odeled a dendrite with active NMDARs that terinates proxially on a soa, deonstrating the phenoenon that I call load-induced bistability, while Korogod & Chernetchenko [46] odeled a syste consisting of two dendrites with NMDARs terinating on a coon soa, deonstrating ultistable behavior. Wolf et al. [47] odeled the specific role of NMDARs in the persistent up and down states observed in the ediu spiny neuron of the nucleus accubens. This paper represents a first step toward a ore general analysis of the electrical effects of NMDARs in concert with other ebrane conductances. I focus on the analysis of stationary current-balance equations in order to characterize instantaneous dynaical regies and exaine the conditions supporting bistability and aplifying behavior. This is done with the recognition that dynaic rather than equilibriu conditions are prevalent in a nervous syste, and that neural state depends on the dynaics of synaptic inputs as well as the kinetics of receptors and ion channels. When NMDARs are involved, any analysis of dynaic behavior is coplicated by the fact that not just neural state, but dynaical regies theselves ay depend on synaptic inputs, and these regies can change on a tie scale coparable to changes in ebrane potential. Dynaics are discussed briefly in the final section of the paper, but analysis of dynaic behavior is beyond its scope. The stationary analysis is nevertheless relevant, because it reveals the requireents for the existence at any instant of bistable dynaics, as well as other characteristics that affect neural behavior under nonequilibriu conditions. For exaple, the presence of two basins of attraction in the state space of a syste can be inferred when stationary analysis indicates the presence of two stable equilibria, and this is relevant even if such equilibria are never reached. In such a syste, sall differences in initial conditions or external perturbations can influence the syste to jup fro one basin to the other, resulting in significantly different paths of evolution in the state space. The analysis herein focuses priarily on the electrical effects associated with the interaction between NMDARs, resting conductance, and synaptically-activated

4 conductances the proxial causes of changes in the electrical state of the neuron fro its resting condition. Of particular significance is interaction with the inward-rectifying potassiu (Kir) channels that are thought to contribute both to the resting potassiu conductance the doinant resting ionic conductance in ost neurons (see, e.g., Wright, [48]) and to the conduction ediated by an iportant class of etabotropic, inhibitory synaptic receptors activated by γ-ainobutyric acid (the GABA B receptors) [49-52]. Because NMDA and non-nmda (i.e., α-aino-3-hydroxy-5-ethyl-4- isoxazolepropionic acid, or AMPA) glutaatergic receptors are often co-located in vertebrate cortical neurons, the iplications with respect to other excitatory synaptic receptors are also considered. Secondary or dependent effects, such as those due to the opening of calciu- and voltage-dependent ion channels are not considered at present, with one iportant exception: the calciu-activated potassiu (SK) channels that are also co-located with glutaatergic synapses in the dendritic spines of pyraidal neurons [53,54]. These channels, like Kir channels, are inward-rectifying [55,56], and they are activated by the calciu conducted into the spine by NMDAR channels. For brevity, I use the abbreviation IRKC to refer to inward-rectifying potassiu channels including both Kir and SK types. During the course of analysis, I exaine a single electrical copartent, and then a neural process (a cable-like dendrite). In analyzing the latter, I apply atheatical results obtained by Gutan and co-workers for dendritic bistability arising fro slow inward calciu currents [57,58]. 1 Methods Neurons were odeled to first order as single electrical copartents, and subsequently as a single-copartent soa coupled electrically to cable-like dendrites. Analysis of stationary versions of the governing equations, which include the instantaneous effects of synaptically-ediated conductances as well as other ion channels, was perfored. The equations were noralized with respect to conductances or resistances and the electrical lengths of the dendrites, although not with respect to ebrane potential since its characteristic values are failiar to physiologists. Nuerical analysis was perfored priarily with SPICE (Siulation Progra with Integrated Circuit Ephasis), a tool that is intended for electronic circuit siulations and is optiized for solution of stiff nonlinear equations. SPICE can solve static equations to obtain dc/equilibriu states. Discrete eleents are described by electrical constitutive relations involving their interconnection points or nodes, and governing equations for interconnected circuits are autoatically generated by application of Kirchhoff s current law (conservation of charge). Built-in linear eleents or user-definable nonlinear voltage-controlled current sources were used for the ipleentation of various ebrane conductances. The product T-SPICE (Tanner Research, Monrovia, CA) was used for the study. In addition, MATLAB (MathWorks, Natick, MA) and Excel (Microsoft, Redond, WA) were used for supporting analysis. 1.1 Models for Mebrane Conductance The acroscopic electrical characteristics of ion channels (i.e., the ean electrical properties of soe large nuber of such channels) were odeled with equations giving

5 current as a function of ebrane potential. Conceptually, the function for each class of channel was separated into two factors: a conductance paraeter and a canonical voltage-dependent function with diensions of volts. Each voltage-dependent function is zero at a unique value of the ebrane potential corresponding to the channel reversal potential, and (except for the NMDA odel) was written in a for in which the reversal potential appears as a paraeter. The conductance paraeter is supposed to account for average conductance and total nuber or density of conducting channels; for synaptically-ediated conductances, it is proportional to the degree of synaptic activation. By convention, the conductance paraeter was equated to the slope of the current-voltage relationship at the reversal potential, and accordingly, the canonical voltage-dependent functions were scaled to unity slope at their reversal potentials. For nonlinear I-V relationships, this convention allows the unabiguous expression of relative conductance as the ratio of conductance paraeters as defined above (although this ratio ay not correspond to epirical values obtained fro local conductance easureents, if the nonlinearity is significant). In analyzing the dependence of neural dynaical regies on synaptically-ediated channels, their instantaneous conductance was typically specified relative to resting ebrane conductance Channel Models Following I derive expressions for the canonical voltage-dependent function characterizing each class of channel odel considered. In addition to NMDARs, these included an ohic odel, a odel based on the Goldan-Hodgkin-Katz (GHK) equations [59,60], and a odel for inward-rectifying potassiu channels. Although the ohic odel is often eployed in coputational neurobiology [61], in reality it is usually an approxiation to the ore physiologically realistic GHK equations. IRKCs are not described well by either odel; their conductance falls off rapidly with increasing ebrane potential for outward currents (i.e., at ebrane potentials above the potassiu reversal potential). Although the particular odel used herein is based on epirical characteristics of Kir channels, it also captures the ost significant conductive characteristic that they have in coon with SK channels the inward-rectifying nonlinearity and thus will be regarded as at least generally representative of SK channels as well. The acroscopic electrical properties of NMDARs were odeled with an expression derived by Jahr and Stevens [2] in which the ean current-voltage relationship in a patch of ebrane takes the for I NMDA = GnbV a[ Mg ] exp(-kv ), (1) where I NMDA is the current, G nb the acroscopic NMDAR conductance in the absence of agnesiu blockade that gives rise to nononotonic behavior, V is ebrane potential, and [Mg 2+ ] the extracellular agnesiu concentration. The constants a and k were assigned the values estiated by Jahr and Stevens, 0.28M -1 and 62V -1 respectively, unless otherwise indicated. A value of [Mg 2+ ] = 1.2M was also assued, so that b a [Mg 2+ ] = With this definition, the canonical voltage-dependent function derived fro the Jahr-Stevens odel can be written

6 f N (V ( 1+ b )V ) = 1 + bexp(-kv The reversal potential associated with this odel is 0V and is oitted as a paraeter. It is iportant to note that the channel conductance paraeter used in conjunction with this odel reflects the slope of the I-V relation under nearly unblocked conditions, and this ust be kept in ind when considering conductance ratios or coparing currents with those of other ion channels particularly when easured near the resting potential, where agnesiu blockade is pronounced. For exaple, in the case of co-located glutaatergic receptors in rat pyraidal neurons, Watt et al. [62] estiate that the ratio of NMDAR to AMPA receptor currents is on the order of 0.4 in iniature excitatory postsynaptic currents. Nuerical analysis shows that, at a ebrane potential V = -70V, the NMDAR conductance paraeter ultiplying the function in (4) would have to be 8 ties larger than AMPA conductance to produce currents in that ratio. (In this analysis, AMPA conductance was treated as ohic with V r0 = 0V.) This is a straightforward consequence of the profound nonlinearity of the NMDAR I-V relation. For ohic ion channels, the voltage dependent for is siply f O (V For channels odeled with the Goldan current equation [59,60], the function was derived fro the expression for ebrane current flux i C carried by a univalent cation C: where [C] in is the internal concentration of C and [C] out its external concentration; and V T is the theral voltage RT/F, where R is the ideal gas constant, T absolute teperature, and F Faraday s constant. By applying the Nernst equation Vr 0 = VT ln( [ C] out /[ C] in ) to introduce the reversal potential for C, and setting the slope of the I-V relationship to unity at V r0, the canonical voltage-dependent function ay ultiately be written f G (V ; V r0 A siilar exercise yields the sae result for a univalent anion. The Goldan current dependence reduces to ohic at V r0 = 0 (i.e., when internal and external concentrations in (4) are equal). Finally, the characteristics of IRKCs were odeled by hand-fitting an ad hoc function to data obtained by Sodickson and Bean [50] for Kir channels in rat CA3 hippocapal neurons. The canonical voltage-dependent for for this channel class is where tanh denotes hyperbolic tangent, d was assigned the value 25V and e the value 0.5 based on the hand fit, and c took the value 13.73V in order to place the zerocrossing of the function at V = V r0. In practice, V r0 is the only free paraeter and is ; Vr0 ) V Vr 0 ). (2) =. (3) V{exp(V / VT )[ C] in [ C] out } ic, (4) exp(v / V ) 1 T VTV{ exp(vr 0 /VT ) 1}{ exp(v /VT ) exp(vr0 /VT )} ) =. (5) V exp(v /V ){ exp(v /V ) 1} f (V ; V r0 r 0 T T d {tanh[(v Vr0 c ) / d ] e } ) =, (6) 1 tanh ( c / d ) Kir r 0 2

7 identified with the reversal potential for potassiu V rk. The fit to Sodickson and Bean s epirical data, with V r0 = -92V and the expression for f Kir above ultiplied by the conductance paraeter 4.4nA/V, is show in Fig. 1. Fig. 1. Fit of an ad hoc canonical voltage-dependent function (in red) to experiental data for the voltage dependence of Kir channel current in rat CA3 hippocapal neurons. The fit for these data is obtained by setting the reversal potential V r0 to the experientally-observed value of -92V, and ultiplying the function f Kir in Equation (6) by the conductance paraeter 4.4nA/V. The data represent the Kir I-V relation in a control case under physiological conditions (i.e., without added Ba 2+ ), and are reproduced with perission fro Sodickson and Bean (1996). Resting Mebrane Model The resting potential in neurons is priarily due to ebrane potassiu conductance, but saller chloride and sodiu conductances also contribute (see, e.g., Wright [48]). When NMDARs are activated in conjunction with the resting conductance alone, the su influence of these other channels deterines the dynaical regies possible. For purposes of illustration, a particular, physiologically plausible resting ebrane odel was constructed and used in siulations of copartents and dendrites. Parallel chloride and sodiu conductances were set to 30% and 4.9% of the total potassiu conductance, respectively, and represented by the Goldan odel. Potassiu conductance was evenly split between Goldan and Kir odels (the forer representing non-rectifying leakage conductance). The coplete odel was cast in the canonical for, as the product of a conductance paraeter G R and a voltage-dependent function f R (V ): f R (V f ) = α [ 0. 5 f G (V ; V rcl Kir (V ; V ) f rk G ) f Chloride and sodiu reversal potentials V rcl and V rna were set to -70V and +60V, respectively, and the potassiu reversal potential V rk to -85V, resulting in a resting potential of -70V. The constant α assued the value in order to achieve unity slope at the resting potential. (V ; V rna G (V )] ; V rk ) (7)

8 1.2 Neural Models In equivalent circuits for a neuron or neural process, the potential of the extracellular space was regarded as the reference potential (ground). Channel odels were configured so as to conduct current fro the intracellular space to an ideal voltage source defined with respect to ground, and representing the reversal potential associated with the particular class of channel. Positive current by convention corresponds to (positive) charge flowing out of a neuron Single Copartent Models For single-copartent odels involving NMDAR and one other class of ion channel, the coplete ebrane I-V relation is written in the for I = GN f N (V ) + G0 f0(v; Vr0 ) + CdV / dt, (8) where I is the ebrane current, V the ebrane potential, the functions f N (V ) and f 0 (V ;V r0 ) are canonical expressions for the voltage dependence of the NMDA and non- NMDA channels, respectively, G N and G 0 are their respective conductance paraeters, and C is ebrane capacitance. For stationary analyses, the dv /dt ter is dropped, and noralization with respect to conductance is accoplished by dividing by G 0 : I / G0 N 0 ; r0 = Γ f (V ) + f (V V ), (9) where Γ is the ratio of the NMDA to non-nmda conductance G N /G 0. In soe single-copartent siulations, ebrane conductance included the resting odel in parallel with NMDAR and additional IRKC conductance. The latter were intended to be representative of either open Kir channels associated with GABA B receptors, or SK channels, and were odeled with the Kir voltage dependence in (6). The agnitude of the additional Kir conductance was specified relative to the resting conductance by a paraeter Κ, and NMDAR conductance was also specified relative to resting conductance by a paraeter Ν. In this case, the governing stationary equation ay be written I / GR = Ν f N (V ) + Κ f Kir (V; VrK ) + f R(V ). (10) Dendritic Models Long, thin dendrites are often odeled with one-diensional cable equations (see, e.g., Rall [63]), with ebrane current paths approxiated as continuous conductances per unit length of dendrite. For concreteness and to liit the degrees of freedo in the analyses described herein, ebrane conductance in all dendrites was assued to include the resting conductance odeled by (7) in parallel with active NMDAR and IRKCs. The full cable equation in this case ay be written

9 2 V 2 x = r a V ν g R f N (V ) + κ g R f Kir (V ; VrK ) + g R f R (V ) + c, (11) t where x is axial position along the dendrite, r a is axial resistance (assued ohic) per unit length, g R is the specific resting ebrane conductance paraeter per unit length, and c is ebrane capacitance per unit length. The paraeter ν is the local ratio of NMDAR to rest conductance per unit length and κ the local ratio of additional Kir to rest conductance per unit length, analogous to Ν and Κ in (10). Mebrane potential V is iplicitly dependent on axial position x. Axial current at any point in the dendrite is equal to -(1/r a ) V / x. As in the single copartent, the V / t ter is dropped fro (11) for stationary analysis. Boundary conditions iposed on a dendrite deterine the for of particular solutions (i.e., V as a function of x) to the stationary version of (11). At the proxial end (x = 0 by convention), this boundary condition is associated with the terination of the dendrite on a soa or another segent of neural process. The dendritic current at the proxial terination, -(1/r a )dv /dx x=0, can be regarded as the output of the dendrite as a whole. By the convention adopted above, it is positive when positive charge flows into the dendrite. For purposes of the analyses herein, the electrical properties of any dendritic segents under consideration were assued to be unifor along the length. In such case, non-diensionalization of the stationary version of (11) with respect to length and resting conductance can be achieved by dividing by r a and g R, yielding 2 V 2 χ = ν f N (V ) + κ f Kir (V ; V rk ) + f R (V ), (12) where χ is a non-diensionalized axial coordinate χ x/λ R, where λ R = ( r a g R ) -1/2 is a conventional electrotonic length constant, equal to the length at which the total axial resistance in the dendrite is equal to the inverse of the total parallel conductance of the resting ebrane. This characteristic resistance value is R R = ( r a /g R ) 1/2. At this point, I consider soe estiates of the absolute conductance and resistance paraeters for an exaple biological dendrite, in order to for an idea of the diensions of a realistic odel and liits on its paraeters. Values of 100Ωc for cytoplasic resistivity and 25µS/c 2 for ebrane specific conductivity are within the physiological range, and with an assued diaeter of 0.5µ, this would give an axial resistance per unit length r a of 5.09MΩ/µ and resting specific ebrane conductance g R of 0.393pS/µ. Lazarewicz et al. [41] developed an estiate of NMDAR density on the dendrites of hippocapal pyraidal neurons based on estiates of receptor density [64,65] and conductance [66] such that at coplete activation the specific NMDAR conductance is 120pS/µ. This is about 305 ties as large as the rest conductance coputed above, providing an absolute upper bound on the paraeter ν, with uch saller values likely in vivo. For ost siulations reported herein, ν assued values in a range about 20, and κ values in a range about 9; these values were selected as representing oderate levels of channel activation, and they support a bistable ebrane characteristic in conjunction with the resting ebrane odel. At κ = 9, the added Kir conductance is 24 ties as large as the resting Kir conductance.

10 For nuerical solution, dendrites were discretized into 100 electrical copartents. Each had its own luped ebrane conductance and axial resistance. Electrical lengths in all siulations were 2.0λ R. A Neuann boundary condition corresponding to a sealed end was ipleented by leaving the distal axial terinal of the ost distal copartent unconnected; Dirichlet (voltage-claped) or Robin (finite proxial load) boundary conditions at the proxial terination were ipleented by connecting the proxial axial terinal of the ost proxial copartent to a voltage source or a odel load, respectively. 1.3 Analysis of Intantaneous Dynaical Regies Bistability and the conditions supporting it were characterized using the discretized versions of the stationary equations (9), (10) and (12). Characterization was accoplished in soe cases by inspection of the stationary ebrane current-voltage relationship for various values of relevant paraeters, allowing identification of unstable as well as stable equilibria, and in soe cases by solving for ebrane voltage or postsynaptic current at various equilibriu states as a paraeter of interest was varied up or down. Such sweeps typically transitioned fro onostable through bistable regies, and the locations of abrupt changes in the ebrane potential corresponding to such transitions could be directly observed. Increental ebrane resistance and gains in onostable regies were characterized by nuerical coputation of the relevant derivatives. In addition, a easure of dendritic current gain was developed by considering the effects on the dendritic output current of an increental test current injected across the dendritic ebrane, uniforly distributed along its length. The NMDAR-induced current gain was defined as the ratio of the change in the output current with NMDAR activation present, to the change induced by the sae test current in the passive dendrite. 2 Results 2.1 Bistability and Aplification in an Electrical Copartent Characteristic Dependence on NMDAR and Non-NMDAR Channels To give a broad idea of the joint dependence of the dynaical states of a neuron on the characteristics of NMDAR and other classes of ion channels, the function f 0 (V ;V r0 ) in equation (9) was identified in separate analyses with the voltage-dependent functions (V -V r0 ), f G (V ;V r0 ), and f Kir (V ;V r0 ) defined for the ohic, Goldan, and Kir odels, respectively, in Section Because only the NMDAR current has a nononotonic voltage dependence, with a single point of inflection, the total ebrane current in such a copartent ay have either one or three zeros but no ore; these correspond respectively to the equilibria associated with onostable and bistable dynaical regies. Results for the NMDAR odel in conjunction with the ohic odel are used here to illustrate the characteristics of these regies. Depicted in Fig. 2 are coputed I-V

11 relationships (corresponding to the results of voltage-clap experients with the voltage varied) for the copartent. Curves were generated for several values of the ratio Γ of NMDAR to non-nmda conductance, and ohic channel reversal potential V r0. In Fig. 2a, for which V r0 = -90V, curve B corresponds to a bistable copartent, while curve A corresponds to a boosting' regie and curve C to a 'self-triggering' regie as defined by Schiller and Schiller [30]. Also shown in Fig. 2b is an equilibriu anifold for this V r0, including the ebrane potentials corresponding to both stable and unstable equilibria, and indicating the liit points of each stable branch of the curve. In the terinology of dynaical systes theory, these correspond to co-diension one saddle-node or fold bifurcations [41,67,68] at which the syste transitions between one and three equilibriu states. Abrupt quasistatic transitions can occur between the stable branches at the liit points, in the directions indicated by the dashed arrows. A fourth regie exists in the paraeter space, in which bistability is not evoked for any value of Γ ; an exaple is characterized by the I-V curves in Fig. 2c, in which V r0 took the value -70V. I refer to such a regie as Γ-onostable, indicating that onostability prevails for any value of NMDAR conductance relative to other ebrane conductances. The curve in Fig. 2d represents (in a sense) a boundary between the first three regies and the fourth; it corresponds to a cusp bifurcation as detailed below.

12 a 60 b Mebrane Current / G 0 (V) A B C Mebrane Potential V (V) c Mebrane Potential V (V) d Relative NMDA receptor conductance Γ Mebrane Current / G 0 (V) D E F Mebrane Current / G 0 (V) Mebrane Potential V (V) Mebrane Potential V (V) Fig. 2. Mebrane I-V relationships and dynaical states of a single electrical copartent. The ebrane contains both NMDAR and ohic conductances. The NMDAR conductance paraeter is expressed by its ratio Γ to the ohic conductance. Panel a depicts the ebrane current (noralized by the ohic conductance) as a function of ebrane potential, for A) Γ = 3; B) Γ = 5; C) Γ = 7; with ohic reversal potential V r0 = -90V. The copartent is onostable for curves A and C and bistable for curve B. Panel b depicts the equilibriu anifold for this syste, i.e., the possible equilibriu potentials as a function of Γ with V r0 = -90V. Heavy curves indicate stable equilibria and the light curve, unstable equilibria. Dashed lines indicate possible transitions between the stable branches at the liit points of the stable equilibria, in the directions indicated by the arrows, and define a hysteresis loop. Panel c depicts ebrane current as a function of ebrane potential, for D) Γ = 1; E) Γ = 3; F) Γ = 5; with ohic reversal potential V r0 = -70V. The copartent is onostable in all cases. Panel d depicts ebrane current as a function of ebrane potential for Γ = 3.56 and V r0 = -78.2V, and corresponds to a cusp bifurcation as detailed in the text. Fig. 3 depicts these regies in the space spanned by the paraeters Γ and V r0, for the Goldan, ohic, and Kir odels in conjunction with NMDAR conductance. The bistable region in each case is bounded by a pair of curves that converge at a point on the right. A 'boosting' regie lies below each pair of curves, and a 'self-triggering' regie above. The point of convergence corresponds to a co-diension two cusp bifurcation, at

13 which two saddle nodes eet and annihilate each another [41,67,68]. The syste is Γ- onostable for non-nmda reversal potentials ore positive that the value at the cusp. As the voltage dependence of the current in the non-nmda odel varies fro superlinear (the Goldan odel) to linear to sublinear (the Kir odel) in character, the point of the cusp oves to ore positive potentials, and the bistable region expands in size. This dependence is of physiological significance: it indicates, for exaple, that a class of ion channel that obeys the GHK equations is capable of inducing bistability in conjunction with NMDAR channels only if its reversal potential is strongly hyperpolarizing. The resting ebrane potential in ost neurons is higher than the values at the cusps in Fig. 3, except for the Kir + NMDAR cobination, and indeed, the exaple resting ebrane odel constructed in Section 0 results in a Γ-onostable copartent in concert with the NMDAR channel odel. Relative NMDA receptor conductance Γ Goldan Ohic Non-NMDA Reversal Potential (V) Kir Fig. 3. Bistable regions of a single electrical copartent. Regions are depicted in the space spanned by the ratio Γ of NMDAR to non-nmda conductance and the reversal potential V r0 of the non-nmda conductance, for three odels for the non-nmda conductance, Goldan, ohic, and Kir odels, as indicated. However, the true values of the paraeters characterizing NMDARs in vivo are not precisely known, and in addition there are several different receptors subtypes [69] (ore than one of which ay be expressed within the sae phenotype) that would be expected to have differing electrical characteristics. In the NMDA odel used to produce Fig. 3 the paraeters estiated by Jahr and Stevens [2] were eployed, with the odel paraeter k set to 62 V -1, but this paraeter has assued values as large as 80V -1 in odeling studies in the literature [70]. Larger values ove the location of the cusp bifurcation to ore positive potentials, i.e., expand the region of bistability. For exaple, for the ohic odel the cusp occurs at V r0 = -78.2V when k = 62V -1, but oves to V r0 = -60.5V when k is set to 80V -1. Siilarly, the resting ebrane odel of Section 0 supports a bistable regie at the higher value of k, although for a liited range of Γ values.

14 2.1.2 Facilitation and Aplification In addition to bistability, aplification is another coputationally significant effect that can be ediated by NMDARs in conjunction with other conductances. As NMDARs are progressively activated in a Γ-onostable regie, the ebrane potential as a function of Γ reaches and passes through a region of relatively rapid depolarization. This nonlinearity can be exploited to achieve coactivation-dependent facilitation, in which there is a superlinear response to the nuber or density of NMDA synapses activated. However, it can also be regarded as setting up an aplifying state, in which current injected by other classes of synaptic receptors has a greater effect on ebrane potential than it would in the resting state. This is reflected by an increased increental ebrane resistance in the high-slope regie. This ode of action is particularly significant given the co-location of NMDA and non-nmda excitatory glutaatergic receptors within the dendritic trees of CNS neurons in fact, in pyraidal cells, at the sae synapses [53,54,71]. Fig. 4a (blue trace) depicts ebrane potential of a single copartent as a function of relative NMDAR conductance for the NMDAR odel in parallel with the resting ebrane odel, and Fig. 4b (blue trace) shows the corresponding increental ebrane resistance relative to the rest state (i.e., ultiplied by the rest conductance).

15 a Mebrane Potential V (V) Κ: b Mebrane Resistance G Κ: Relative NMDA receptor conductance Ν Fig. 4. Mebrane potential and increental ebrane resistance in a single electrical copartent. Each is depicted as a function of Ν, the ratio of NMDAR to resting conductance. The odel contains additional Kir conductance beyond that in the resting ebrane, paraeterized by its ratio Κ to the rest conductance, and eant to represent the effects of activated Kir or SK channels. Mebrane potential is shown in Panel a, and increental ebrane resistance, noralized by ultiplication by the rest conductance, in Panel b. The blue curves are for NMDARs in conjunction with rest conductance alone; the copartent is onostable for the blue and black curves, while the red curves correspond to a cusp bifurcation. The Role of Inward-Rectifying Potassiu Channels With regard to NMDAR-induced bistability and aplification, it is clear that potassiu conductance also plays a significant role; increasing the relative agnitude of any of the other ion channel conductances considered would push the cusp bifurcation of the syste to ore negative values of the copartent equilibriu potential. The results

16 for the Kir odel deonstrate that increasing IRKC conductance has the reverse effect and for this reason, the co-location of either GABA B synaptic receptors or active SK channels with NMDARs in a neuron is a subject of particular interest. As expected, siulations of a copartent with parallel resting and active conductances deonstrate that an increase in inward-rectifying potassiu conductance can induce bistability in an otherwise onostable copartent, and then expand the range of NMDAR activation over which bistability prevails. Such an increase also affects the aplifying characteristics of a copartent when in a Γ-onostable regie. In addition to the case of NMDARs acting in concert with rest conductance alone (Κ = 0), Fig. 4 depicts how the dependence of ebrane potential and increental ebrane resistance on NMDAR activation (i.e., Ν) varies with IRKC activation (paraeterized by Κ). For the black traces in Fig. 4, the syste is Γ- onostable; the red traces depict ebrane potential and increental resistance as the syste reaches a cusp bifurcation, which occurs at Κ = As Κ increases, the effect is inhibitory in the sense that the copartent hyperpolarizes for low levels of NMDAR activation, and the steep part of relation between V and Ν is pushed to higher values of Ν. However, in the vicinity of that steep region, the gain with respect to injected currents (as quantified by the ebrane resistance) draatically increases with IRKC activation, and the syste is capable of powerful aplification of sall changes in input, whether these occur at NMDA or other classes of synapses. This effect is due siply to the interaction between the nonlinearities of the NMDAR and IRKC I-V relations. A liit is reached at the cusp bifurcation, where the ebrane resistance becoes infinite as a discontinuity appears in the relationship between V and Ν. Coactivation of GABA B synapses or SK channels with NMDARs thus provides a potentially powerful coputational echanis for facilitation, aplification, and shortter eory functions. SK channels are found in intiate association with glutaatergic synapses in dendritic spines, and are (in a sense) autoatically activated, at least in part by calciu influx through NMDARs theselves. GABA B synapses, conversely, provide a echanis for independent control of facilitation and aplification. Pharacological evidence suggests that, like NMDARs, GABA B receptors are distributed in the dendritic trees of CNS neurons in which they occur [72]. 2.2 Bistability and Aplification in a Neural Process Characteristics of Dendritic Bistability and Aplification Consider next the dynaical regies of a neural process, i.e., a dendrite, odeled by the stationary equation (12). When the ebrane current per unit area is a nononotonic function of ebrane potential V, there is potential for bistable behavior; the axial resistance as well as ebrane properties ay play a role in inducing such behavior. By varying the relative agnitudes of the active conductances in (12) over physiologically reasonable ranges, a variety of ebrane I-V characteristics can be generated, including onotonic, nononotonic (N-shaped) with a single zero, and nononotonic with three zeros. Although this paper is concerned with NMDARs, any of the results discussed below are relevant whenever a nononotonic ebrane I-V characteristic prevails. The

17 resultant dynaical regies are rather ore coplicated in a dendrite than a single copartent. The theory of the bistable dendrite has been advanced to a considerable degree by Gutan and coworkers [57,58,73,74]. The dendritic odel they considered also coprised a one-diensional cable equation with spatially unifor conductive properties, and with a bistable ebrane I-V characteristic. The physiological basis of this was the presence of slow, persistent inward calciu currents, but the theory can be applied equally well to bistability associated with NMDAR channels. Gutan [57,58] considered the stationary boundary-value proble in which the dendrite is sealed (i.e., allows no current flow) at the distal end (resulting in a Neuann boundary condition), and is voltage-claped at the proxial end (a Dirichlet boundary condition). Several iportant theoretical results obtained for this odel [57] can be suarized as follows: 1. Below a critical dendritic length, the solution V (χ) of the stationary equation is unique for all values of the proxial clap voltage, and the relationship between the current flowing into the dendrite and the clap voltage is single-valued; 2. For dendrites longer than this critical length, the solution is not unique for soe range of values of the clap voltage. The current flowing into the dendrite ay assue any of two or ore values for any clap voltage within this range. Furtherore, this voltage range increases as the length of the dendrite increases; 3. The nuber of possible solutions within this voltage range also increases as dendritic length increases, but all except two of these solutions are unstable. The stable solutions each feature a onotonic dependence of V on χ, with the potential at the distal end approaching either the largest or sallest zero-crossing voltage of the ebrane I-V relation. There are thus two possible stable values of current into the dendrite for any clap voltage within the range. For brevity, I refer to such pairs of stable solutions as, respectively, the high and low states. To illustrate the character of these solutions for a dendrite with active NMDARs, Fig. 5a depicts siulation results for current flowing into a dendrite as a function of the clap voltage for the odel (12), for paraeter values ν = 20 and κ = 9, and with dendritic length Λ in non-diensionalized coordinates as a paraeter. Fig. 5b shows exaples of the ebrane potential in the dendrite as a function of χ, for a dendrite long enough to support bistability. The clap voltage is a paraeter. Gutan and coworkers [57,75] defined dendritic bistability in the narrow sense described above: as the existence of two possible stable states, with two different currents into the dendrite, when the process is voltage-claped proxially. I refer to this as Dirichlet bistability. The voltage-clap condition is not entirely relevant to in vivo physiology, but can be regarded as defining the stationary current-voltage relation for the dendrite as a whole, and fro this point of view Dirichlet bistability is an intrinsic property of a dendrite, which in the present case is dependent on its instantaneous state of NMDAR and IRKC activation.

18 a Output Current R R (V) Λ: b Proxial Clap Voltage (V) Mebrane Potential V (V) Axial Position / λ R Fig. 5. Output characteristics of a dendrite with active NMDARs under a proxial Dirichlet boundary condition. Panel a depicts the dendritic output current (noralized by ultiplying by the characteristic resistance R R of a dendrite of unity electrotonic length) as a function of the clap voltage defining the proxial boundary condition. Electrotonic length Λ of the dendrite is a paraeter, taking the values indicated. Purple curves correspond to onostable regies; blue curves indicate the high state and red the low state, in bistable regies. Panel b shows possible distributions of ebrane potential as a function of axial position in a dendrite with length (Λ =1.4) long enough to support bistability. The paraeter is the proxial clap voltage, which takes the value of the ebrane potential at the y-intercept of each curve or pair of curves. Again onostable solutions are indicated by purple curves and bistable solutions by blue (the high state) and red (the low state) curves. For given conductance paraeters, the critical length at which Dirichlet bistability appears ay be approxiated [74]: 1 / 2 crit ( π / 2 )( g in / g R ) Λ, (13) where g in is the iniu value of the slope of the (nononotonic) dendritic ebrane I-V relation per unit length, and g R the specific resting ebrane conductance paraeter. When expressed in absolute units, the critical length L crit = λ R Λ crit (π /2)(-g in r a ) -1/2 is

19 independent of the choice of the rest state as reference, as expected fro physical considerations. The quantity g in /g R corresponds to the iniu value of d/dv [ν f N (V ) + κ f Kir (V ;V rk ) + f R (V )]. I exained Dirichlet bistability in dendrites with active NMDAR and inwardrectifying potassiu conductance using (12) in conjunction with a sealed-end distal boundary condition. The expression in (13) was found to give good agreeent with critical lengths obtained by nuerical solutions of (12), so long as a bistable ebrane I- V characteristic was present. Interestingly, however, I found that Dirichlet bistability can occur even when the ebrane I-V relationship has only a single zero, so long as that relationship is nononotonic. Under these conditions, the two possible values of the output current are of the sae sign. I refer to this phenoenon as load-induced bistability, which can be seen to arise due to the presence of axial resistance in the structure and current flow induced by the proxial clap. Dirichlet bistability, whether load-induced or not, is associated with the eergence in soe distal portion of the dendrite of a regie in which the relationship between an injected current and the local ebrane potential is N-shaped with three zeros. Load-induced bistability can occur with activation of NMDARs only (i.e., with κ = 0 in (12)), but because the axial resistance is ohic, bistable regies exist only for hyperpolarized proxial voltages (with the NMDAR odel used in this study). This behavior irrors that of the single copartent with parallel NMDAR and ohic ebrane conductance, in which bistable regies exist only for ohic reversal potentials below typical resting potentials. In the context of a dendrite that is part of a neuron, the voltage-clap boundary condition corresponds to an assuption that the dendrite has essentially no electrical influence on the soa and/or reainder of the neuron. Realistically, however, the reainder of a neuron ust present a finite conductance (referred to hereafter as the proxial load ), which ay be large copared to the dendritic conductance if the dendrite terinates on or near the soa, or ay be of siilar agnitude if a length of passive neural process intervenes. Generally, as the conductance of this load becoes saller (assuing it is passive in nature), the critical length for dendritic bistability correspondingly becoes saller for a given density of active synapses. In any such case, the relevant boundary-value proble is deterined by the condition that the current into the dendrite ust atch the current flowing out of the load. This results in a Robintype boundary condition in which both the ebrane potential and its spatial derivative appear. Via this boundary condition, the dynaical state of the dendrite is affected by the conductance and instantaneous reversal potential of the proxial load. (This reversal potential is, by definition, the value of the potential at the boundary for which no current flows fro the reainder of the neuron.) The overall current-voltage relation for the dendrite alone, derived fro the voltage-clap analysis, is still conceptually useful in such cases: the dendrite ay be regarded as an electrical black box in parallel with the load conductance, and when the I-V relationship of the two taken together has three zeros, the dendrite is bistable in cobination with the load. A set of exaples is shown in Fig. 6, in which this technique has been used to visualize the variety of regies that can prevail in a dendrite governed by (12) in conjunction with a proxial Robin boundary condition. Paraeters were chosen so that the dendritic ebrane I-V relationship was nononotonic in each case, and the I-V relationship of the proxial load was linearized about its instantaneous reversal potential