David HORN and Irit OPHER. School of Physics and Astronomy. Raymond and Beverly Sackler Faculty of Exact Sciences

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1 Cmplex Dynamics f Neurnal Threshlds David HORN and Irit OPHER Schl f Physics and Astrnmy Raymnd and Beverly Sackler Faculty f Exact Sciences Tel Aviv University, Tel Aviv 69978, Israel Abstract We study an integrate and re mdel f an adapting neurn. Using tw dynamic threshlds we accunt fr cmplex lng term behavir f single neurns under peridic pulsed inputs. We nd that the pattern f tempral respnse is that f a staircase f inter-spike interval values. The characteristic time scales f transitins between these ISI values are f the rder f tens f secnds. The tempral behavir f ur system can be described as a path thrugh the 2-dimensinal phase space f dynamic threshlds. 1 Intrductin Single neurns display interesting cmplex behavir fr dierent inputs. When driven by a cnstant supra-threshld current, a neurn will usually re regularly with sme frequency. After a while, this frequency will decrease t a steady state value manifesting adaptatin f the neurn t the cnstantly applied stimulus. A ttally dierent tempral behavir is seen when the neurn is driven by a peridic stimulus. In a recent experiment, Tal et al: [7] studied the eect f 2 minutes applicatin f 2ms current pulses, using dierent frequencies. The current amplitude was just abve threshld, sucient fr making the neurn re at 1 Hz when stimulated with 1 Hz pulsed inputs. This behavir we call a "1:1 respnse". When the input frequency was higher, abut 1/2 f the neurns respnded with cmplex ring patterns, and did nt display this simple 1:1 respnse. Examples f ISI series recrded in respnse t stimulatin f 3 and 30 Hz are shwn in Fig. 1. It is evident that a high frequency input causes the neurn t respnd with a staircase f ISI values. The time Crrespnding authr. Phne Fax Submitted fr ral presentatin at CNS99. 1

2 Input Frequency=3 Hz Input Frequency=30 Hz Figure 1: ISI recrded in respnse t stimulatins f 3 Hz (upper frame) and 30 Hz (lwer frame). All ISI values are integer multiples f the input perid. intervals between successive spikes grw, but the ring rate des nt necessarily reach a steady state value. In this paper we present a minimal mdel that describes such cmplex and variable behavir. It is based n dynamic threshlds assciated with an I&F neurn. 2 Dynamical Threshlds in an Integrate-and-Fire Neurn An Integrate-and-re (I&F ) neurn [3] can be dened by the frmula C dv dt =?g L(V? V rest ) + I (1) where V is the membrane ptential, C is the capacitance, g L is the leak cnductance and I is the external input current. When V reaches threshld a spike is emitted and V is reset t V reset such that V rest < V reset < V threshld V reset + : (2) Refractriness may be added as a shrt dead-time after ring. 2

3 When the external current has a cnstant cmpnent that is much larger than statistical uctuatins, ne can apprximate the inter-spike-interval by ISI =? C g L lg(1? g L I ef f ): (3) where I ef f = I? g L (V reset? V rest ). It fllws then that the ring rate will be I ef f =C fr large input currents. MacGregr[5, 6] discusses the use f dynamical changes f the threshld parameter as suggested by varius authrs since the early 30s. One natural extensin [2] is using the threshld t describe adaptatin, r fatigue, accunting fr the fact that the ring rate decreases in time fr a given cnstant input. This can be dne by hypthesizing a behavir f d dt =?(? 0) (4) with an updating prescriptin such that after every a spike is emitted! + : (5) A typical value fr is in the range f msec, which is the range in which dierent types f neurns display adaptatin. Alternatively ne can intrduce a `current threshld', a term which appears naturally if we try t accunt fr after-hyperplarizatin eects by mdifying the I&F equatin: C dv dt =?g L(V? V rest ) + I? g x x(v? V x ) (6) x dentes a variable like the cncentratin f Ca ++ in the cell. In that case g x = g AHP and V x = V K, the K + reversal ptential. This current threshld is linearly dependent n x and has als fatigue type eects n the ring rate f the neurn. x is increased after every spike and is endwed with decay dynamics f the type x! x + x (7) x dx dt where nw x determines the tempral scale f adaptatin. =?x; (8) This prblem was recently investigated in detail by Liu and Wang [4] whse ntatin we have adpted in the descriptin s far. They pint ut that the AHP current leads t adaptatin, and can therefre act as the equivalent f the vltage-threshld. They cmpare the tw in light f experimental ndings by Ahmed et al: [1] and cnclude that the current-threshld serves as a better descriptin. 3

4 3 Mdel f Tw Dynamical Threshlds The mdel we prpse fr qualitative descriptin f the data[7] discussed in the Intrductin relies n using bth types f threshlds tgether. In ding s we assume that there is a new behavir assciated with time scales f tens f secnds, that has t perate in additin t the cnventinal adaptatin represented by either the vltage-threshld r the current-threshld terms. Our descriptin is therefre represented by the set f equatins 4-8, where 6 describes the membrane ptential and all the ther the behavir f bth dynamical threshlds. The interplay f bth threshlds gives us the richness f many dierent ISI patterns fr a given high-frequency input. We have run simulatins with the 2ms pulse prtcl, using dierent input frequencies. The structures we btain can be characterized as a staircase f ISI values. This is because the dynamics f the threshlds change slwly and the system stays fr lng times at metastable values f the m : n rati between input and utput frequencies, as can be seen in Fig. 2a. Such a staircase is the backbne f the ISI structure bserved experimentally, as seen in Fig. 1. Once nise is added t the system, as in Fig. 2b, the results can be made t t the data bserved experimentally. 4

5 150 (a) (b) Figure 2: Examples f staircase f ISI values fr a 30 Hz stimulus in ur mdel: a. In the absence f nise, the system stays fr rather lng times in xed m : n ratis f input t utput frequencies. The ratis 1:1, 2:1, 3:1 and 4:1 are metastable. At later times we bserve mre cmplex behavir f scillatins between the ratis f 4:1 and 5:1. b. When little nise is added t the input, we get time series that resemble experimental results. The basic staircase structure can still be seen, but the pattern is less regular. Using ur simple mdel we were able t mimic the behavir f many neurns. In each case, the distributin f ISI as well as the tempral behavir f the neurn were successfully recnstructed. An example is shwn in Fig. 3, resembling the tempral pattern f the neurn shwn in the tp frame f Fig Phase Space Analysis The dynamics f ur cmbined mdel can be separated int fast and slw cmpnents. The tempral integratin that the neurn perfrms during the 2 ms current pulse is the fast cmpnent. The dynamics f the threshlds, whse values vary cnsiderably nly after a large number f spikes have been emitted, is the slw cmpnent. This separatin allws us t calculate, fr each pint (; x), the expected m : n rati 5

6 Figure 3: Simulatin results that resemble the experimental example f 3 Hz shwn in Fig. 1. The similarity between the tw time series is evident. Furthermre, statistical prperties f these tw series are the same. between input and utput frequencies. It fllws that we can cnstruct a tw-dimensinal phase space where the tempral develpment f the system is represented as a trajectry. An example that crrespnds t the ISI staircase f Fig. 2a is shwn in Fig. 4. X cmplex behavir 5:1 4:1 3:1 2:1 Figure 4: A tw dimensinal phase space f ur system. Different areas in this phase space crrespnd t dierent m : n ratis between input and utput frequencies. The circles represent the trajectry crrespnding t the ISI time series f Fig. 2a. 1:1 θ This representatin pens the pssibility fr an investigatin f the behavir f the same neurn under input currents f dierent frequencies. S far we have allwed urselves the freedm f chsing apprpriate parameters fr dierent runs even fr the same neurn. We hpe t btain a cnsistent descriptin that will allw us t t the dierent runs n the same phase space, and nd the crrect dynamics that will cnsistently accunt fr all f the behavir displayed by the same neurn. 6

7 Acknwledgment It is a pleasure t thank Shimn Marm fr sharing his data with us and fr many helpful cnversatins. References [1] B. Ahmed, J.C. Andersn, R.J. Duglas, K.A.C. Martin and D. Whitteridge, Estimates f the net excitatry currents evked by visual stimulatin f identied neurns in cat visual crtex. Cerebral Crtex submitted. [2] D. Hrn and M. Usher, Neural Netwrks with Dynamical Threshlds. Phys. Rev. A40, [3] B. W. Knight, Dynamics f encding in a ppulatin f neurns. J. f Gen. Physil. 59, [4] Y.-H. Liu and X.-J. Wang, Adaptatin and decrrelatin in crtical pyramidal neurns: A generalized integrate-and-re mdel with stchastic inputs. t be published in Behaviral/Systems Neurscience. [5] R. J. MacGregr, Neural Mdeling, Plenum Press. [6] R. J. MacGregr, Neural and Brain Mdeling, Academic Press. [7] D. Tal, E. Jacbsn, V. Lyakhv and S. Marm, Frequency-tuning f a single neurn transfer functin. Preprint. 7

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