OUTPUT STREAM OF BINDING NEURON WITH DELAYED FEEDBACK

Transcription

1 binding neuron, biological and medical cybernetic, interpike interval ditribution, complex ytem, cognition and ytem Alexander VIDYBIDA OUTPUT STREAM OF BINDING NEURON WITH DELAYED FEEDBACK A binding neuron (BN) with delayed feedback i conidered. The neuron i fed externally with a Poion tream of intenity λ. The neuron' output pike are fed into it input with time delay. The reulting output tream of the BN i not Poionian, and we look for it interpike interval (ISI) ditribution. For BN with threhold an exact mathematical expreion a function of λ, and BN' internal memory, τ i derived for the ISI ditribution, and for higher threhold it i found numerically. The ditribution found are characterized with dicontinuitie of jump type, and include ingularity of Dirac' δ -function type. It i concluded that delayed feedback preence can radically alter neuronal output firing tatitic. 1. INTRODUCTION The role of input pike timing in functioning of either ingle neuron, or neural net ha been addreed many time, a it contitute one of the main problem in neural coding. The role of timing wa oberved in procee of perception [9], memory [6], object binding and/or egmentation [3]. At the ame time, where doe the timing come from initially? In reality, ome timing can be inherited from the external world during primary enory reception. In auditory ytem, thi happen for the evident reaon that the phyical ignal, the air preure time coure, itelf ha pronounced temporal tructure in the milliecond time cale, which i retained to a great extent in the inner hair cell output [,4,1]. In olfaction, the phyical ignal i produced by mean of adorption-deorption of odour molecule, which i driven by Brownian motion. In thi cae, the primary enory ignal can be repreented a Poion tream, thu ha not any remarkable temporal tructure. Neverthele, temporal tructure can appear in the output of a neuron fed by an untructured ignal. After primary reception, the output of correponding receptor cell i further proceed in primary enory pathway, and then in higher brain area. During thi proceing, tatitic of pottimulu piking activity undergoe ubtantial tranformation (ee, e.g. [4]). After thee tranformation, the eventual activity i far away from the initial one. Thi proce i cloely related to the information condenation [7]. We now put a quetion: What kind of phyical mechanim might underlie thee tranformation? It eem that, among other, the following feature are reponible for piking tatitic of a neuron in a network: (i) everal input pike are neceary for a neuron from a higher brain area to fire an output pike (ee, e.g. [1]); (ii) a neural net ha numerou interconnection, 9

2 which bring about feedback and reverberating dynamic in the net. Due to (i) a neuron mut integrate over a time interval in order to gather enough input impule to fire. A a reult, in contrat to Poion tream, the hortet ISI between output pike will no longer be the mot probable. Thi wa oberved long ago [11] in numerical experiment with leaky integrate-and-fire (LIF) neuronal model and confirmed recently in exact mathematical derivation for binding neuron [15]. Due to reverberation, an individual neuron' output impule can have ome delayed influence on the input of that ame neuron. Thi can be the ource of poitive feedback which reult in etablihing of dynamic partially independent of the timulating input (compare with [7]), and which govern neuronal piking tatitic. In thi paper, we conider a implet poibility to tet influence of (i), (ii), above on neuronal firing tatitic. A neuronal model we take the binding neuron (BN) one. Exact mathematical expreion i derived for output ISI ditribution a a function of input Poion tream intenity, λ, BN' internal memory, τ, delay value in the feedback line,, when BN ha threhold. For higher threhold the ditribution are calculated numerically, by mean of Monte Carlo algorithm. The ditribution found are characterized with dicontinuitie of jump type, and include ingularity of Dirac δ-function type. It i concluded that delayed feedback preence can radically alter neuronal output firing tatitic. feedback, N threhold Σ output tream input tream τ memory Figure 1. Binding neuron with feedback (ee [14] for detail). denote delay time in the feedback line. τ i imilar to the tolerance interval dicued in [8, p. 4]. Multiple input line with Poion tream are joined into a ingle one here.. METHODS For analytical calculation we conider threhold value N =..1. BN WITHOUT FEEDBACK The binding neuron model [14] i inpired by numerical imulation of Hodgkin-Huxley-type point neuron [13], a well a by the leaky integrate-and-fire (LIF) model [11]. In the binding neuron, the trace of an input i remembered for a fixed period of time after which it diappear completely. Thi i in the contrat with the above two model, where the potynaptic potential decay exponentially and can be forgotten only after triggering. The finitene of memory in the binding neuron allow one to contruct fat recurrent network for computer modelling a well a obtain exact mathematical concluion concerning firing tatitic of BN. Recently, [15], the finitene i utilized for exact mathematical decription of the output tochatic proce if the binding neuron i driven with the Poion input tream. 93

3 The BN work a follow (ee Fig. 1 with the delay line removed). Each input impule i tored in the BN for a fixed period of time, τ, and then i forgotten. When the number of tored impule, Σ, become equal or bigger then the BN' threhold, N, the BN fire output pike, clear it internal memory, and i ready to receive freh input. Normally, any neuron ha a number of input line. If input tream in each line i Poion, all line can be joined into a ingle one, like in Fig. 1, with intenity, λ, equal to um of intenitie in the individual line. Recently [15], the output tatitic wa calculated for thi model with N =. In thi work we will need the following exact expreion from [15]. ISI ditribution probability denity function, P () t, where t denote the output ISI duration, ( m+ 1) ( m+ 1) λ ( t mτ) mτ t ( m+ 1) τ P ( t) dt = e λdt+ ( m + 1)! k λ k k + e (( t ( k 1) τ) ( t kτ) ) λdt, m=,1, K, k! l k m (1) and the firt moment, W of the ditribution (1), 1 W1 tp t dt 1 1 () = +, λ e λτ 1 ().. FIXED FEEDBACK LINE Any output impule of BN with feedback line (BNF) may be produced either with impule from the line involved, or not. We aume that, jut after firing and ending output impule, the line i never empty. Thi aumption i elf-evident for output impule produced without impule from the line, or if the impule from the line wa involved, but entered empty neuron. In the letter cae, the econd (triggering) impule come from the Poion tream, neuron fire and output impule goe out a well a enter the empty line. On the other hand, if impule from the line trigger BN, which already keep one impule from the input tream, it may be quetionable if the output impule i able to enter the line, which wa jut filled with the impule. We expect it doe. Thi mean biologically that we ignore the refraction time - a hort period neceary for a nervou fibre to recover from conducting previou pike before it i able to erve for the next one. Thu, at the beginning of any output ISI, the line keep impule with time to live, where ]; ]. It i clear, that variability of the input Poion tream hould be combined with the variability in value in order to calculate the output tream propertie, like ISI probability denity P ( t). In thi ubection, we define an auxiliary probability denity, P () t, in which the time to live of the impule in the feedback line i put equal to a fixed value at the beginning of any output ISI. Thu, intead of conidering a tationary firing proce in which both firing moment and are determined by the input Poion proce, we conider a proce in which, after each firing, the line keep impule with time to live equal to the ame value ] ; ]. In order to derive P ( t) it i uitable to eparate poible value t of ISI duration into everal group a hown in Fig.. 94

4 Figure. Domain of t ued for calculating P () t. In the cae C1, t<. Here output impule mut be triggered without the line impule involved. Therefore, the ditribution for uch ISI value i the ame a for BN without feedback: () (),. P t = P t t < (3) Conider the cae C. The probability to obtain ISI exactly equal to i not infiniteimally mall. Thi event i equivalent to the event A () S that BN tart empty at moment and appear without 1 triggering in tate S (keep impule) at moment. In order to obtain the probability PA { ( ) }, 1 let u take into account that P ()d can be obtained a the product of PA { ( )} and the probability to get input impule in infiniteimal interval d, which i λ d. Therefore, S 1 S 1 P () PA { S ( )} =, (4) 1 λ which, together with Eq. (1), give the δ-function' ma in the expreion for P ( t) at point t =. In order to keep expreion horter, let u aume that < τ, and calculate ISI ditribution for the cae C3, above. Due to the aumption made, the probability to obtain ISI value < t + τ i jut equal to the probability that firt input impule come at required moment t. Therefore, t P () t = e λ λ, < t + τ. (5) Conider cae C4, t + τ. It i realied if three independent event occur in erie: (i) A () S ; (ii) interval ] ; + τ [ i free from input impule; (iii) BN without feedback tart from tate S at moment + τ and i firtly triggered at moment t. Thee event are independent ince their realization are defined by behaviour of Poion input tream on interval, which are mutually dijoint. Due to the aumption made, the probability to have both (i) and (ii) i the ame a to have in the Poion input tream an ISI longer then + τ, and (iii) ha the probability P ( t τ ) dt. Thu, P t = e P t t + (6) λτ ( + ) () ( τ ) τ. Equation (3), (4), (5), (6) can be written together a um of ingular and regular part: r P () t = P () t + P (), t (7) where P () t = e λ λδ( t ), (8) 95

5 e tλ, t ]; ], (**) r P () t = λe, t ]; + τ], (*), + λτ ( + ) e P ( t τ), τ t, (*) (9) where aumption.3. DERIVATION OUTLINE < τ i taken into account. When initial data i forgotten, the firing proce of BN with delayed feedback become tationary. Thi bring about a tationary ditribution, f ( ), for time to live, ]; ], of an impule in the feedback line at the moment of beginning of any output ISI. Having exact expreion for f ( ), one could calculate required output ISI ditribution a follow: P () t = P () t f( ) d. (1) In order to find f ( ), conider the tranition probabilitie P ( '),, ' ]; ], which give probability that at the beginning of ome output ISI, the line ha impule with time to live, provided that at the beginning of the previou ISI it had impule with time to live '. P ( ') can be found baed on known expreion for P ( t). the following equation: f ( ) i then found a normalized to 1 olution to P ( ') f( ') d' = f( ). (11) 3. MAIN CALCULATION 3.1. TRANSITION PROBABILITIES From the meaning of P ( t) it follow that Eq. (9)(**) allow to calculate P ( ') for P e λ ( ' ) ( ') = λ ( ' ), < ' ]; ]. < ' : (1) Eq. (8) and (9)(*) decribe ituation when one ISI tart with impule in the feedback line, which ha time to live equal, and the next ISI tart with impule in the line, which ha time to live equal. Thu, P ( ') ha ingularity of δ -function type at =. For calculating it ma, one hould take (8), (9)(*) with replaced with ' and calculate integral over admiible value of t : ' + τ λ ' λ( τ+ ') λ' λ' e λ' + e λdt+ e P ( t ' τ) dt = λ' e + e. ' ' + τ 96

6 Here we ue P () t dt = 1. Thu, P ( ') i the um of two function where P ( ') = P (, ') + P(, '), (13) 1 λ ( ' ) e λ < ( ' ), ' ]; ] P1 (, ') =,, ' P e e λ' λ' (, ') = δ( )( λ ' + ). The tranition probability P ( ') 3.. DELAYS DISTRIBUTION i normalized: P ( ') d= 1. Here we found probability denity ditribution f ( ). For thi purpoe let u repreent f ( ) a λ f ( ) = aδ( ) + g( ) = aδ( ) + e ϕ ( ), (14) where a i a dimenionle contant, and g ( ), ϕ ( ) are ordinary function. After ubtituting (13) and (14) into Eq. (11), and eparating term without δ -function, one obtain ae ( ) + ( ' ) ( ') d ' = ( λ λ λ ϕ ϕ ). Thi equation can be eaily olved with repect to ϕ ( ), which deliver g ( ) a aλ ( ) g ( e λ ) () = 1. (15) Now take into account that f ( ) mut be normalized: a + g() d = 1, which give for a : a = λ 4e λ + e + λ ( 3) 1. (16) 3.3. ISI DISTRIBUTION For calculating P ( t) ubtitute (7), (8), (9) and (14), (15), (16) into Eq. (1). Thi give 97

7 t r P () t e λ r = λt( aδ( t ) + g()) t + ap () t + P () t g() d. (17) Further tranformation of (17) depend on the t value. Baic domain of t are hown in Fig. 3. Figure 3. Domain of t ued for calculating integral in (17). Conider cae A. Here integration domain hould be plit into two with the point = t. Thi give t () = λ () + λ + λ ( ) + λ ( ), P t e tgt a te e gd te gd which after tranformation become ( t ( + 7) + 1 ( + 1) ) λ λ λ e λ λte λt e λ t e λ P () t =, t <. (18) λ λ ( + 3) e + 1 At t =, ISI ditribution P ( t) ha δ -function type ingularity: t λ 4λe P () t = δ ( t ), t ] ε; + ε[. λ ( λ + 3) e + 1 (19) Conider cae B. Here integration in (17) can be performed over the entire domain ]; [ uniformly, which give P () t = e λ f( ) d = e λ, < t < τ. () Conider cae C. Here integration domain hould be plit into two with the point = t τ, and Eq. (17) turn into the following: λτ ( + ) () = ( τ ) ( ) + λ ( ) +. λ P t e P t g d e g d ae Here in the firt integral, ( t τ ) [; ] [; ] [; τ ]. Thi allow to identify from Eq. (1) λ( t τ) exact expreion for P ( t τ ), which i e λ ( t τ ): After tranformation, one obtain () = λ ( τ) ( ) + λ ( ) +. λ P t e t g d e g d ae 98

8 λ( ) ( K + K1t+ Kt + e ) λe P () t =, τ < t <+ τ, λ e (4λ+ 6) + (1) where K λ = (λτ + 4λτ+ 4λ+ 6) e λτ+ 1, K e K e λ λ 1 = ( 4 (1 + λτ )) λ, = λ. Conider cae D. Here Eq. (17) turn into the following: λτ ( + ) λτ ( +) () = ( τ ) ( ) + ( τ ). P t e P t g d ae P t Let u introduce a new variable of integration, u = t τ : λ( t u) λ( τ+) () = ( ) ( τ ) + ( τ ), P t e P u g t u du ae P t () t τ From thi expreion we ee, that for calculating the integral one need to ue Eq. (1) either with ingle, or with two conecutive value of the parameter m. Namely, if for ome m : mτ t τ < ( m+ 1) τ, then one hould ubtitute term from (1), correponding to that m intead of P ( u) in the (). In the oppoite ituation, there exit uch an m, that mτ < t τ < ( m+ 1) τ <. In thi cae, domain of integration in the Eq. () hould be plit with point ( m + 1) τ, and a P ( u) one hould ubtitute term from (1), correponding either to m, or to m + 1. Thu, when t [ +τ ; [, then all poible ituation are parameterized with the above mentioned number m in uch a way that if t [ + ( m+ 1) τ ;( m+ ) τ[, then ue term from (1) with that m, and if t [( m+ ) τ ; + ( m + ) τ[, then plit integration domain and ue term with both m, and m +1. For example, if t + [ τ ; τ[, then m = and () turn into λ( t u) λu () = λ ( τ ) + λ ( τ), P t e e ug t u du ae t t τ which give after tranformation λ 1 ( λ + 6 λ + 1) e P () t = λ ( t τ) e + λe, λ e (4λ+ 6) + t [ + τ,[. τ (3) Graph of P ( t) i hown in Fig

9 Figure 4. Example of ISI probability denity function, calculated in accordance with Eq. (18), (19), (), (1), (3), left panel, and numerically, by mean of Monte Carlo method, right panel. For both panel: τ = 1 m, =8 m, λ = 1-1. In the left panel, N =, in the right one, N = 4. Curve found numerically for N = fit perfectly with one hown in the left panel. In the numerical experiment 36 pike were produced OUTPUT INTENSITY Let u found mean output ISI, W. Output intenity i invere of W. The W i defined a Ue here Eq. (1): W = tp () t dt. d W = t dt P () t f () d = d f () tp () t dt. Ue here repreentation (8), (9) and Eq. (): + τ λ W = d f() t e λ dt+ e λ + tλe dt + λτ ( + ) d f () e tp ( t τ ) dt + = λ λ( τ+ ) (1 + λe ) (1 + λτ + λe ) = d f () + λ d f e λ e 1 + τ d λτ ( + ) () τ. λτ Ue here (14), (15), (16), which give after tranformation: W λ λτ ( λ+ e + λe ) ( 1) = λ λ e λ ( + + 3)(1 ). (4) λτ e 3

10 The output intenity i then λ = 1/W. At large input rate the following relation take place λ 1 lim λ =. λ (5) 4. NUMERICAL SIMULATION In order to check correctne of obtained analytical expreion, a well a to obtain an impreion how ISI ditribution look like for higher threhold, a C++ program wa developed, which directly model functioning of BN with delayed feedback. The Poion input tream were generated by tranformation of uniformly ditributed equence of random number (ee, e.g. Eq. (1.14) in [1]). The ISI probability denity i found by counting output ISI of different duration and normalization. In the program, ditribution of time to live of impule in the feedback line wa calculated a well. Numerically obtained curve fit perfectly with the analytical expreion for P () t given in Eq. (18), (19), (), (1), (3), and for f ( ) given in Eq. (14), (15), (16). 5. CONCLUSIONS AND DISCUSSION It i well known [5] that neuronal ytem with noie and delayed connection can demontrate very complicated behaviour, which i hardly ubjected to rigorou mathematical analyi. In thi paper, we conider a implet verion of uch a ytem: the binding neuron with delayed feedback, which i timulated with Poion tochatic tream. For thi ytem, we calculate exact expreion for the output ISI probability denity function if BN ha threhold. For threhold and 4, the ISI ditribution i alo found numerically. The function obtained have remarkable peculiaritie, which ugget what could happen with piking tatitic of individual neuron in elaborated network with delayed connection. For threhold we alo found the output intenity a a function of the input one. The limiting relation (5) can be undertood a follow. At moderate timulation ome input pike are lot without influencing output due to high probability of long input ISI. At high intenity every two conecutive input impule trigger the BN and end impule into the feedback line, provided it i empty. Thu, output intenity hould be λ / plu firing, caued by additional timulation from the line. Thi additional timulation ha maximum rate equal to1/, which explain (5). Acknowledgment. I thank to A. Andrew for ending me the D. MacKay' paper. During preparation of thi paper the following free oftware were ued: (i) Linux operating ytem; (ii) computer algebra ytem ``Maxima'' ( REFERENCES [1] ANDERSEN P., RAASTAD M., STORM J.F., Excitatory ynaptic integration in hippocampal pyramid and dentate granule cell, in Cold Spring Harbor Sympoia on Quantitative Biology 81-86, (Cold Spring Harbor Laboratory Pre, Cold Spring Harbor, 199) [] CARIANI P., Temporal code, timing net, and muic perception, J New Muic Reearch, Vol. 3, 1, pp [3] ECKHORN R., BAUER R., JORDAN W., BROSCH M., KRUSE W., MUNK M., REITBOECK H.J., Coherent ocillation: a mechanim for feature linking in the viual cortex?, Biol Cybern, Vol. 6, 1988, pp

11 [4] EGGERMONT J.J., Rate and ynchronization meaure of periodicity coding in cat primary auditory cortex, Hearing Reearch, Vol. 56, 1991, [5] HAKEN H., Brain Dynamic, Springer, Berlin 8. [6] HEBB D.O., The Organization of Behaviour, Wiley, New York [7] KÖNIG P., KRÜGER N., Symbol a elf-emergent entitie in an optimization proce of feature extraction and prediction, Biol Cybern, Vol. 94, 6, pp [8] MACKAY D.M., Self-organization in the time domain, in Self-Organizing Sytem, edited by M.C. Yovitt, G.T. Jacobi, G.D. Goldtein (Spartan Book, Wahington), 196, pp [9] MACLEOD K., BÄCKER A., LAURENT G., Who read temporal information contained acro ynchronized and ocillatory pike train?, Nature, Vol. 395, 1998, pp [1] MOORE B.C.J., Coding of ound in the auditory ytem and it relevance to ignal proceing and coding in cochlear implant, Otology & Neurotology, Vol. 4, 3, pp [11] SEGUNDO J.P., PERKEL D., WYMAN H., HEGSTAD H., MOORE G.P., Input-output relation in computerimulated nerve cell, Kybernetic 4, 1968, pp [1] SMITH G.D., Modelling the Stochatic Gating of Ion Channel, in Computational Cell Biology, edited by Ch.P. Fall, E.S. Marland, J.M. Wagnerand, J.J. Tyon, Springer, Singapore,, pp [13] VIDYBIDA A.K., Neuron a time coherence dicriminator, Biol. Cybern, Vol. 74, 1996, pp [14] VIDYBIDA A.K., Inhibition a binding controller at the ingle neuron level, BioSytem, Vol. 48, 1998, [15] VIDYBIDA A.K., Input-output relation in binding neuron, BioSytem, Vol. 89, 7,