Noise Effects on Two Excitable Systems: Semiconductor Laser and Neuron Network Oscillations

Transcription

1 Noie Effect on Two Excitable Sytem: Semiconductor Laer and Neuron Network Ocillation Athena Pan, Wei-Mien Mendy Hu Department of Phyic, Univerity of California, San Diego I. Introduction An excitable ytem conit of three general tate: ret tate, excited (or firing) tate, and a refractory (or recovery) tate. When unperturbed, the ytem tay at it ret tate. Under mall perturbation, the ytem perform mall-amplitude linear ocillation around it ret tate. However, for large enough perturbation, the ytem can be driven out of it ret tate and goe through the excited and recovery tate before returning to it ret tate [Linder]. Such repone i highly nonlinear. There exit many excitable ytem in nature. In thi paper, we tudy two pecific ytem in detail: emiconductor laer with optical feedback and neuron network ocillation. We decribe the theorie ued to model thee two excitable ytem and explain how noie can be introduced to induce coherence reonance in thee ytem. II. Correlated Noie Induced Optical Coherence Reonance

2 Fig. A. Experimental etup of a emiconductor laer with optical feedback [Lindner]. Semiconductor laer ubjected to external optical feedback (Fig. A) can exhibit dynamical intability [10, 8]. Specifically, when the injection current i kept above, but cloe to, the olitary laer threhold, irregular dropout in laer intenity can be oberved [5]. Giacomelli et al ha further hown experimentally that the coherence (or regularity) in the laer intenity dropout can be tuned by changing the amount of input noie in the pump current. Coherence reonance can be oberved for an optimal amount of noie, which i neither too large nor too mall Fig. B [5]. I(t) Fig. B. Temporal evolution of the intenity of a emiconductor laer with optical feedback. The noie level i (a) dbm/mhz, (b) -5.5 dbm/mhz, (c) dbm/mhz [5]. 1. Lang-Kobayahi Model

3 The dynamic of emiconductor laer with optical feedback can be modeled with the Lang-Kobayahi (LK) model [7]. The LK model aume the emiconductor laer i under ingle-mode operation and that multiple reflection from the external feedback mirror can be ignored due to low reflectivity. Under ingle-mode, ingle reflection condition, we need only to add an extra term to the tandard emiconductor laer rate equation to account for the feedback field [7, 4, 9]: de( t) 1 i G( E, N) E( t) exp[ i ] E( t ) F ( t (1) dt dn dt ), CN N( t) G( E, N) E( t). () e The material-gain function can be expreed a follow: g N( t) N0 G. (3) th 1 E( t) The definition of all variable ued above i ummarized in table 1. The random fluctuation due to pontaneou emiion i repreented by F E (t), which i Guaian white noie with zero mean and correlated in time (i.e. F ( t) F ( t) R ( t t) ). The teady tate olution of the LK equation are of the following form [8, 10]: E( t) P exp( it ), N( t) N th n, (4) where P E(t), n i the carrier denity change, and repreent the frequency hift from the olitary laer frequency. Subtituting equation (4) into (1), eparating the real and imaginary part, we obtain the following two equation: G G E, N) co( ), (5) ( f G( E, N) in( ) ( co( ) in( )). (6) Variable Definition E (t) Complex electric field envelope inide the laer N (t) Exce carrier number Linewidth enhancement factor Solitary laer frequency, e Invere of photon and carrier lifetime, repectively Feedback trength / rate, related to facet and external reflectivity [8] Time it take for light to travel to the external mirror and back F E (t) Langevin force Gauian white noie due to pontaneou emiion C Pump rate, directly related to pump current N th Threhold carrier number N 0 g Table 1. Definition of variable in rate equation. f f f f E f E E f

4 Combining equation (3), (4), (5), and (6), we ee that elliptical equation [8]: and n are related by an gn gn 1. (6) That i the multiple olution to the LK equation lie on an ellipe in the (, n ) phae pace. Fig. C how a poible et of olution in phae pace. The characteritic equation of the Jacobian matrix of the LK equation are rather complicated [11]. However, it can be hown that olution on the upper half of the ellipe, denoted by tar, are untable addle point, uually referred to a antimode. The diamond in the center of the ellipe correpond to the teady tate olution in the abence of optical feedback. Solution at the lower half of the ellipe, referred to a mode, may or may not be table depending on many parameter one of them being the feedback trength. The olution indicated by an arrow in Fig. C repreent the maximum-gain mode (MGM). Operating under thi mode, the laer i mot efficient outputting maximum power with minimum carrier denity change n. Phyically, it correpond to optimal contructive interference between the intracavity field and the feedback field [8]. Fig. C. Steady tate olution (fixed point) of the LK equation in (, n ) phae pace [8]. For mall feedback, the mode are table and the laer operate at a frequency very cloe to it olitary frequency,. However, a the feedback trength increae, many of the mode alo become untable and the ytem attempt to move toward the MGM. However, before the ytem can reach the MGM, it pae by an antimode,

5 which "throw" the ytem back toward the center of the ellipe. Mulet et al numerically imulated the phae pace trajectory for a emiconductor laer biaed cloe to it threhold and ubjected to optical feedback (Fig. D) [9]. A hown in Fig. D, a the ytem travel toward the MGM from other untable mode (diamond), it come too cloe to an antimode (croe), which drive the ytem toward the center of the ellipe, reulting in a dramatic increae in ( t) ( t) ( t f ) and a intenity dropout. Here, (t) i the difference between the phae of the intra-cavity field (t) and that of the feedback field t ). ( f Fig. D. Evolution of the trajectory in phae pace of a emiconductor laer with optical feedback and pumped cloe to threhold [9].. Optical Coherence Reonance By tuning the noie characteritic in the pump current, Buldu et al howed that the coherence in the intenity dropout can be optimized [4]. To model the pump current noie, an extra term, (t), wa added to equation (): dn dt C [ 1 ( t)] N N( t) G( E, N) E( t). (7) e th The ytem evolve in time cale on the order of ten of picoecond, which i fater or of the order of the characteritic time cale of the fatet electrical modulation that can be introduced experimentally [6]. Therefore, the pump current noie cannot be adequately modeled with Gauian white noie. Intead, Buldu et al adapted

6 correlated noie of the Orntein-Uhlenbeck type, which i Gauianly ditributed with zero mean and correlation of form: t) ( t) D t t exp c c (, (8) where c i the correlation time of the noie. One can olve equation (1), (7) and (8) numerically and recover the intenity dropout or phae jump in the time erie of the laer output (Fig. E) [4]. Fig. E. Simulated intenity and phae difference a a function of time. A peak in the phae difference between intra-cavity field and feedback field correpond to a dropout in the intenity [4]. D The noie level can be tuned in the imulation by changing the variance c. Fig. F how the phae jump a a function of time for three different value of. The phae jump are mot coherent (or regular) for moderate amount of noie (Fig. F(b)), which correpond to perturbing the ytem in reonance. That i the average perturbation time interval i cloe to the ytem' intrinic recovery time. For low noie level (Fig. F(a)), the perturbation i rarely large enough to drive the ytem out of it table mode. Therefore, the time interval between ucceive jump in (t) are relatively large and le regular. For very high level noie (Fig. F(c)), the ytem i frequently driven out of it current mode before it ha enough time to recover back to it table mode. That i the average perturbation time interval i much horter than the ytem' recovery time. Therefore, we ee that the peak in (t) broaden and the time interval between ucceive peak are irregular.

7 Fig. F. The coherence behavior of (t) a the noie level change: (a) , 1 (b) , (c) Noie correlation time i kept contant at [4]. c 4p Interetingly, coherence reonance can alo be oberved by fixing the noie D level and changing the noie correlation time. Fig. G(b) how that the c c peak in the phae (t) i mot coherent for moderate correlation time c 57. 6p [4]. Short correlation time correpond to high frequency modulation in the pump current. In thi high frequency limit, the carrier dynamic in the emiconductor act a a lowpa filter, preventing the ytem from reponding to the high frequency modulation. Therefore, the interval between ucceive peak in Fig. G(a) are large and irregular. On the other hand, in the low frequency limit, or long correlation time, the carrier have enough time to repond and the ytem follow the noie modulation in the pump current. The incoherence in the noie naturally tranlate into the incoherence of the ytem output (Fig. G(c)). Only for moderate correlation time will the ytem repond in reonance.

8 Fig. G. Coherence behavior of phae difference (t) a the noie correlation time change: (a) p, (b) 57. p, (c) 153. p. Noie intenity i kept c c 6 c contant at [4]. III. Noie-Induced Network Ocillation in a Reduced Burting Model 1. Reonant Integrate-and-fire Model In thi article, the author reduced the Hindmarh-Roe (HR) model by modifying the integrate -andfire (IF) model with a reet mechanim after a threhold croing, a given by the two linear ODE and the reet rule: when reache The reemble a voltage-like variable while decribe the low dynamic of an adapting current, imilar to the lowly relaxing variable of HR model. The tationary tate i at while no input ( ) i applied. The firt ODE behave a integrate-and-fire model: when the voltage-like reache the threhold, i reet to and meanwhile an action potential (pike) occur, or more pecifically aying, a harp voltage change acro the nerve membrane. To mimic the dynamic of HR model that a pike make the increae of, will reet to whenever threhold croing happen.

9 The four parameter A, B, C and D in RIF model are choen to fit it impedance magnitude with that of HR model [S. Reinker, 006], o a to mimic the ubthrehold behavior of HR model it exhibit a ubthrehold reonance to ignal with input period near 336 time unit [Reinker, 003]. The fitting reult given in the article are A= 0.03, B= 1.358, C = and D= Reult and Dicuion The Determinitic Firing in RIF Model Determinitic cae with different et of,, and are hown in Figure 1. When, the RIF ytem eventually evolve to equilibrium tate. A contant.4 applied, the ytem tart to cro threhold and fire regularly, and the firing frequency increae with the increae of the input intenity. Mathematically, with any contant input the table tate i given a 1. However, with the threhold-croing-and-firing mechanim in RIF model only when will the ytem go to the table tate, otherwie it tart to firing periodically. Fig. 1 The determinitic RIF model: (a) It goe to table tate when. (b) A, the ytem can not continuou the periodic firing. The preence of the firt firing depend on the initial condition of at. A long a i negative and i mall enough, the would likely to be poitive that make reach the threhold in the early time tep. (c)-(g) how the regular 1, the ame a defined in the Appendix.

10 firing pattern with different et of,, and. (c)&(d) The firing rate increae with for that it i related with the increaing rate of. (e) Lowering the to 0, the time coure of inhibitory-to-firing i reduced uch that the period of the firing pattern lightly decreae. On the other hand, the period will be prolonged if (f) i et to larger value 1 or (g) threhold i lightly lifted the to (h) While i ufficiently mall (0.01) that after each firing the could till be poitive, then it enter endle burting phae. The Stochatic Firing in RIF Model Adding Gauian white noie into RIF model, we get the two-dimenional tochatic differential equation which decribe Orntein-Uhlenbeck proce when reache, where denote the Wiener proce and the contant denote noie intenity. Figure how the tochatic FIR cae both with but different noie intenity. Interetingly, the noie not only can kick to reach the threhold but alo give rie to have multiple threhold croing the burt firing. Becaue after a firing the reet value of i till cloe to the threhold, with ufficiently large noie it highly poible to to cro threhold again. However, each threhold croing make increae by, and the larger the i the more inhibition exert on. Therefore, burting phae will be terminate at ome time and entering the recover period. Fig. The tochatic FIR with noie level (a) and (b), both demontrate noie-induced multiple threhold croing (burt). The number of burt in the ame burting phae can be tell form the increment of value becaue we know each croing make increae by. In (a) around, uddenly jump to indicating there are three time croing, a hown in the inet. Note that a the noie level increae, not only the firing rate lightly increae but the number of burt in the hort time period ditinctly increae a well. Network behavior of RIF Model To invetigate the network propertie of the RIF neuron, the implet way i to bring an all-to-all coupling with an identical coupling trength into the tochatic differential equation:

11 when reache The ummation in the firt SDE model the ynaptic coupling between N neuron by etting that the voltage of th neuron will be lifted by the amount of ( ) whenever the th neuron ( ) fire. In other word, any one of the neuron in the network croing the threhold will make all the other neuron jump by. Here we chooe to be poitive o neuron are gradually puhed to threhold for each kick. Figure 3 diplay the rater plot of the coupled 50 RIF neuron with different et of noie intenity and coupling trength. Figure 3 (a)-(c) the coupling i fixed but with different noie intenity repectively. When noie level i low ( ) the firing happen occaionally and i eemingly random. For intermediate noie ( ) it demontrate a ynchronized and periodically firing among the network, but for ufficiently high noie ( ) though the neuron till fire rapidly the ynchronized activity diappear. Meanwhile, coupling trength alo ha influence on thi kind of tochatic ocillation, a hown in Figure 3 (d)-(f). With mall noie intenity, the network ynchronization can only occur when the coupling trength i trong enough Figure 3 (e), and perit even with larger (f). The only difference between (e) and (f) i that it take a while to tart ynchronize for a lower coupling trength. In a more complex neural network model, except for having the poitive coupling, ome portion of neuron will be et to have a negative on the other o a to decribe the inhibitory network behavior.

12 Fig. 3 (a)-(c) Rater plot of imulation of 50 RIF neuron [S. Reinker, 006], with different noie level (a: 0.08, b:0.1, c:0.90) but with fixed coupling trength Synchronized tochatic network ocillation occur at intermediate noie trength and are table over a wide range of value (between approximately 0.1 and 0.85). (d)-(f) The imulation with the dependence on coupling trength (d: 0.06, e:0.10, f:0.0) but fixed noie level The ynchronized ocillation appear when the coupling i trong enough. 3. Stochatic Analyi of RIF Neural Network During the burt recovery period, the ytem i decribed by two linear tochatic differential equation Eq. (3) and (4), from which we can obtain it aociated Fokker-Plank equation (FPE). By olving FPE we will get the probability ditribution of the ytem that help u to clarify: 1) the ytem behavior during the burt recovery phae; ) approximate the probabilitie of threhold croing, and 3) the expected number of neuron croing threhold in the contrained ytem. Following the procedure hown in Appendix, we got the uncontraint 3 probability denity of one neuron, which i a Gauian ditribution 3 The probability denity can go over the threhold. Even it not

13 where and are the center of the Gauian (the firt moment of and ), and are the variance which i proportional to (noie intenity), and i the correlation coefficient. Since threhold only exit in variable, it i convenient to conider the marginal probability denity for Alo, the quantity i define a the probability that > in the uncontrained and uncoupled ytem at time t, Figure 4 (a) and (b) how the time evolution of for a ingle neuron with different noie level. A we ee the move toward the table point and arrive there roughly at 300 to 350. From Appendix, we know the variance of,, i proportional to. Therefore, a centered at, the fraction of that lie outide of threhold i - dependent. The larger the i, the larger the portion of the tail of fall in the range >, hown in Figure 4 (c), and the higher the probability to have threhold croing. And thi threhold-croing probability ha been defined a and hown in Figure (d). Fig. 4 (a) and (b) how at t=100, 150, 00, 50, 300, and 350 with the noie level 0.1 and 0.0, repectively. (c) with different noie intenity. The portion of the tail of the denity that exceed i directly related to the variance of the ditribution, which i proportional to the quare of the noie intenity ( ). (d) the larget ha the dramatically increae in while for mall i very cloe to 0, denoting that the high lead to the high probability of threhold croing at the mot likely time.

14 Another quantity, defined a the average number of neuron that cro the threhold pontaneouly at a given time interval of length, i given a where i the average of over the period to. Form figure 4 (d) we know reache maximum at and almot tay a a contant afterward. Therefore indicate how likely the iolated neuron are to cro threhold at time. For different noie level, their value at are hown in Table 1. Table 1 The larger noie level ha the larger, i.e., the threhold croing i more likely to happen with higher noie. With the quantitie and, we can explain how a mall number of firing neuron can kick the ytem to burting phae. The mechanim of coupling, lifting the voltage by whenever one neuron fire, i equivalent to hift the center of to approach the by the amount of, i.e.,. Thi hift in make it right tail enter the range >, and increae, a hown in figure 5. For the cae with lightly larger hown in figure 3 (e), the ynchronization doen t exit until time 000. Obviouly, with mall the majority of neuron may have to be puhed multiple time o a to hift their ufficiently, which increae to higher probability. Thu, in the beginning the firing pattern i eemly random due to jut few of the neuron originally cloed to threhold can cro the threhold. However, when thee neuron fire the ret will alo be puhed to approach the threhold. A the voltage kicking goe on, majority of neuron become more and more cloed to the threhold. Thu, whenever a neuron fire, thoe that are cloet to the threhold will alo be puh to fire. Then the ret will alo be puhed by the previou firing event, o on and o forth, jut like a threhold croing avalanche that firing trigger the fire of the other neuron in the hort time period. Fig. 5 (a) how the probability (olid), and (dahed) of a ingle neuron,before and after a neuron firing ( ). (b) the correponding before and after the hift of, with different noie level. IV. Appendix: Solving Probability Denity

15 The probability denity equation 4, for the olution of the uncontrained tochatic differential i a Gauian, and can be determined by olving the aociated Fokker-Plank Equation: which include four drift term, and the only one diffuion term owing to the noie input in the equation. The initial condition at an initial time are given by Here, we are intereted in the evolution of jut after an action potential generate, o we take initial condition at a the reet time following an action potential, and et and a well. If we expre by it Fourier tran-form with repect to x and y, i.e., by we obtain the Fourier tranform of Eq.(??) by replacing with and (or with (or ), with initial condition. Since we already know that i a Gauian and therefore make the anatz that i a Gauian a well, and the olution i where and ( ) are moment and variance. Inert thi anatz in to Eq. (A.4) and get the equation require that and mut atify the following ODE And then we follow the approache hown in (Riken, 1989), calculated and with Mathematica, and eventually we get 4 The can be larger than the threhold becaue FPE i not involved in the reetting mechanim of and, which i applied when imulating the two SDE Eq. (3) and (4).

16 where i the eigenvalue of matrix The probability denity i Fig. A1 The plot of,, and. V. Reference 1. B Lindner et al., Noie effect in the excitable ytem, Phy. Report 39, (004).. S. Reinker et al., Noie-Induced Coherence and Network Ocillation in a Reduced Burting Model, Bulletin of Math. Biology 68:6, (006).

17 3. S. Reinker et al., Reonance and Noie in a Stochatic Hindmarh Roe Model of Thalamic Neuron, Bulletin of Math. Biology 65, (003). 4. J. M. Buldú et al., Effect of external noie correlation in optical coherence reonance, Phy. Rev. E 64, (001) 5. G. Giacomelli, Phy. Rev. Lett. 84, 398 (000). 6. I. Ficher, Phy. Rev. Lett. 76, 0 (1996). 7. R. Lang, K. Kobayahi, IEEE J. Quantum Electron. QE-16, 347 (1980). 8. G.H.M. van Tartwijk, G.P. Agrawal, Prog. Quantum Electron., 43-1 (1998). 9. J. Mulet and C. R. Mirao, Phy. Rev. E 59, 5400 (1999). 10. C.H. Henry, R.F. Kazarinov, IEEE J. Quantum Electron. QE-, 94 (1986). 11. G.H.M. van Tartwijk, D. Lentra, Quantum Semicla. Opt. 7, 87 (1995).