d dt S = (t)si d dt R = (t)i d dt I = (t)si (t)i

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2 d d S = ()SI d d I = ()SI ()I d d R = ()I

3 d d S = ()SI μs + fi + hr d d I = + ()SI (μ + + f + ())I d d R = ()I (μ + h)r

4 d d P(S,I,) = ()(S +1)(I 1)P(S +1, I 1, ) +()(I +1)P(S,I +1, ) (()SI + ()I)P(S,I,)

5 d d P(n,) = (n 1, )P(n 1, ) + μ(n +1, )P(n +1, ) [ (n,) + μ(n,) ]P(n,) (n,) = ()n μ(n,) = μ()n

6 d d ()exp () [ M() ] g(z,) = ( ()z μ() )(z 1) z g(z,) dz d d d ( ) ()exp () M() = ( ()z μ() )( z 1) = ( () μ() ) + () ( [ ]) d = () = C () = ()d M() = μ()d

7 g(z,) = f V () z 1 W () V () g(z,) = f = z n z 1 [ V () W ()]z +V() +W () V() g(z,) = V () +W () W ()z ( [ ]) V () = exp () M() W () = ()V ()d n o p(,) =1 V () (n = ) p(n,) V () +W () ( ) = W () n 1 V ()V () V () +W () ( ) n +1 (n )

8 n 2 () = V () n() = V () V () V () 2W () +V () V() V () F = 2 n () n() 2W () +V() V () = V ()

10 1..5 y x

11 .2.1 y x

13 d d P(n,) = ()P(n 1, ) ()P(n,) P(n,) = ()n n! f ISI () = ()exp(()) exp( () ) (Rae Code) (Time Code) f ISI () = f ( )g()d f ( ) = () 1 exp() g() = = B(,) () 1 exp() 1 () = ()d Fano Facor =1(always) ( + ) + (Tsubo, [3]; Ikeda [4]) Ge good fi ISI in vivo Fracal-renewal process conradics synfire chain/ bursing fracional-gaussian-noise-driven Poisson process (fgndp)

15 Memory funcion ype Langevin equion d d A = ia ( s)a(s)ds + f () () f () f ()AA * 1 [6] H. Mori, Progr. Theor. Phys. 33 (1965) 423 Convoluion-less ype Langevin equaion d d A = i() ( )A() + g() () = (s)ds () = g()g * ()AA * 1 [7] M. Tokuyama and H. Mori, Progr. Theor. Phys. 55 (1976) 411.

16 d (i) Memory funcion ype Maser equaion [8] j [ ] d P(n,) = d K nj( )P( j,) K ( )P(n,) jn (ii) Non-saionary (Time-convoluion less) ype Maser equaion [9] d d P(n,) = j [ L ] ()P( j,) L ()P(n,) nj jn

18 w ISI () = exp ( ) Coefficien of Variabiliy (CV) CV = Var() E[] Local Coefficien of Variabiliy (LV) LV = 1 3( i i+1 ) 2 =1 m 1 ( i + i+1 ) 2 Fano FacorF Allan Facor (A) p(n,t) = (T)n n! exp( T) F(T) = Var[n] E[n] =1 A(T) = E[(n k +1 (T) n k (T))2 ] E[n k (T)] m1 i=1.. Index of clusering =1 Variance-o-mean raio = 2F(T) F(2T) =1

19 w ISI () = 1 () CV = 1 exp ( ) LV = Mixed Gamma disribuiondoubly Gamma process p ISI () = = d = w ISI (,;) f ()d 1 () ( + ) ()() exp( ) () 1 exp( ) 1 ( + ) + (in viro) (Ikeda [3]) (Bea funcion of The second kind) (Tsubo [4])

20 Number disribuion for Gamma Process is esimaed by p(n ;,T) = P(n,T /) P((n +1),T /) P(, x) = 1 () x 1 exp()d Mixed Gamma Disribuion semi-parameric model p(n F;,T) = b a p(n ;,T)dF() Incremen is no independen: i is desirable o avoid for analyzing sequences wih synfire chain/bursing

21 d d P(n,) = ()P(n 1, ) ()P(n,) P(n,) = ()n n! () = exp( () ) ()d E[n] = Var[n] = () F(T) =1 (always)

22 () () () E[n] = Var[n] = () W () =1 exp(()) exp( ) 1 exp( ) ( ) 1 exp 1 exp( ) ( ) 1+ 1 ln 1+ ( ) F(T) =1 (always) 1 (1 + ) 1 exp

23 f ISI () = ()exp () ( ) () = 1+ E[] = 1 Var[] = f ISI () = ( 1+ ) 1+ ( ) 2 ( 2 ) CV = Generalized Pareo disribuion Var[] E[] = 2 () = 1 f ISI () = 1 exp E[] = 1/ 1+ 1 Var[] = 2/ >1 (always) Weibull disribuion CV =

24 d d P(n,) = [ ( )P(n 1,) ( )P(n,) ] d

25 d d ( ) P(n,) = () [ (n 1) + ]P(n 1, ) [ n + ]P(n,) Feller (1963) [1] () = () = 1+, =1

26 p(n,) = n, n = p(n,) = ( exp(()) ) n exp(()) n ( ) n () = ()d

28 () () = ()d () = 1+ E[n] = exp(()) 1 [ ] Var[n] = exp(()) [ exp(()) 1] Fano Facor Allan Facor F(T) = exp( (T) )>1 A(T) = 2F(T) F(2T)

29 (i) (ii) () () = ()d exp( ) ( 1 exp( ) ) 1+ ln( 1+ ) F() = exp(()) exp 1 exp( ) (1 + ) ( ) (iii) 1 exp The second iem corresponds exacly he fracional power law In he Fano Facor

30 (In he limi of for he generalized Polya process) p(n,) = n,1 Soluion for, Geomeric Disribuion p(n,) = ( exp(()) )1 ( exp(()) ) n n =1 Mean and Variance E[n] = exp(()) [ ] Var[n] = exp(()) exp(()) 1 [ ] Fano Facor F(T) = exp( (T) )1> This is conradic he feaure of Fano Facor of experimens in he region of small T

31 p(n,) = exp( (1) )(1) n F() = L n ( ) L n (x) = Laguerre'sPolynomial E[n] = exp(()) + [ exp(()) 1] Var[n] = exp(()) [ 2 exp(()) 1]+ [ exp(()) 1] Fano Facor (of signal averaging) n p(n ) = n exp(()) [ 2 exp(()) 1]+ [ exp(()) 1] exp(()) + [ exp(()) 1] n! exp( )

32 wihin Pr(d P,n) = n P nd (1 P ) d d f (P r n,s n ) = Hence, one obains SDT disribuion 1 1 B(r n,s n ) P r 1 n (1 P ) s n 1 Pr(d n,r n,s n ) = Pr(d P,n) f (P r n,s n )dp = n B(n d + r,d + s ) n n d B(r n,s n )

33 6.8 spike duraion ime (SDT) disribuion Pr(d n,r n,s n ) = n B(n d + r,d + s ) n n d B(r n,s n ) (r n,s n ) = (1.2,3.2) (r n,s n ) = (2.3,.6) (r n,s n ) = (.5,.6)

34 (Omied here)

35 A(T) T A A value

36 From Lowen and Teich Fracal-Based Poin Processes (Weiley, NY, 25)

37 (iii) (ii) (i) cf. Table 6.4

38 () = f f ( 1 exp[ f ] )+ s log( 1+ s ) s FF() = exp( () ) Fas mode (conribue o decrease FF value) + Slow mode (conribue o fracional power law) s s =.5

Spiral Ganglion neuron Type-I neuron (fas) Type-II neuron (slow) are no found

39 LIBERMAN MC. Audiory-nerve response from cas raised in a low noise chamber. J. Acous. Soc. Am. 63(2): , Spiral Ganglion neuron Type-I neuron (fas) Type-II neuron (slow) are no found in 1978 Idenified by negaive binomial disribuion + Poisson disribuion he negaive binomial disribuion Pr(n) Number n

40 () = 1+

42 T Ts

43 ISI disribuion Parameer of exponenial disribuion is subjeced o Gamma disribuion f () = e () e 1 d = Pareo disribuion B(,1) 1 +1 ( + ) Spike-couning disribuion Parameer of Poisson couning process is subjeced o Gamma disribuion Negaive binomial disribuion P(N() = n) = ( + n 1)! n!( 1)! + n +

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