{"title": "Response Analysis of Neuronal Population with Synaptic Depression", "book": "Advances in Neural Information Processing Systems", "page_first": 523, "page_last": 530, "abstract": null, "full_text": "Response Analysis of Neuronal Population with Synaptic Depression\n\nWentao Huang Institute of Intelligent Information Processing, Xidian University, Xi'an 710071, China wthuang@mail.xidian.edu.cn Shan Tan Institute of Intelligent Information Processing, Xidian University, Xi'an 710071, China shtan@mail.xidian.edu.cn\n\nLicheng Jiao Institute of Intelligent Information Processing, Xidian University, Xi'an 710071, China lchjiao@mail.xidian.edu.cn Maoguo Gong Institute of Intelligent Information Processing, Xidian University, Xi'an 710071, China mggong@mail.xidian.edu.cn\n\nAbstract\nIn this paper, we aim at analyzing the characteristic of neuronal population responses to instantaneous or time-dependent inputs and the role of synapses in neural information processing. We have derived an evolution equation of the membrane potential density function with synaptic depression, and obtain the formulas for analytic computing the response of instantaneous re rate. Through a technical analysis, we arrive at several signi cant conclusions: The background inputs play an important role in information processing and act as a switch betwee temporal integration and coincidence detection. the role of synapses can be regarded as a spatio-temporal lter; it is important in neural information processing for the spatial distribution of synapses and the spatial and temporal relation of inputs. The instantaneous input frequency can affect the response amplitude and phase delay.\n\n1\n\nIntroduction\n\nNoise has an important impact on information processing of the nervous system in vivo. It is signi cance for us to study the stimulus-and-response behavior of neuronal populations, especially to transients or time-dependent inputs in noisy environment, viz. given this stochastic environment, the neuronal output is typically characterized by the instantaneous ring rate. It has come in for a great deal of attention in recent years[1-4]. Moreover, it is revealed recently that synapses have a more active role in information processing[5-7]. The synapses are highly dynamic and show use-dependent plasticity over a wide range of time scales. Synaptic short-term depression is one of the most common expressions of plasticity. At synapses with this type of modulation, pre-synaptic activity produces a decrease in synaptic. The present work is concerned with the processes underlying investigating the collectivity dynamics of neuronal population with synaptic depression and\n\n\f\nthe instantaneous response to time-dependence inputs. First, we deduce a one-dimension Fokker-Planck (FP) equation via reducing the high-dimension FP equations. Then, we derive the stationary solution and the response of instantaneous re rate from it. Finally, the models are analyzed and discussed in theory and some conclusions are presented.\n\n2\n2.1\n\nModels and Methods\nSingle Neuron Models and Density Evolution Equations\n\nOur approach is based on the integrate-and- re(IF) neurons. The population density based on the integrate-and- re neuronal model is low-dimensional and thus can be computed ef ciently, although the approach could be generalized to other neuron models. It is completely characterized by its membrane potential below threshold. Details of the generation of an action potential above the threshold are ignored. Synaptic and external inputs are summed until it reaches a threshold where a spike is emitted. The general form of the dynamics of the membrane potential v in IF model can be written as v dv (t) = dt v(t) + Se (t) + v\nN X\n\nJk (t) (t\n\nk=1\n\ntsp ); k\n\n(1)\n\nwhere 0 v 1, v is the membrane time constant, Se (t) is an external current directly injected in the neuron, N is the number of synaptic connections, tsp is occurring time of k the ring of a presynaptic neuron k and obeys a Poisson distribution with mean k, Jk (t) is the ef cacy of synapse k . The transmembrane potential, v , has been normalized so that v = 0 marks the rest state, and v = 1 the threshold for ring. When the latter is achieved, v is reset to zero. Jk (t) = ADk (t), where A is a constant representing the absolute synaptic ef cacy corresponding to the maximal postsynaptic response obtained if all the synaptic resources are released at once, and Dk (t) act in accordance with complex dynamics rule. We use the phenomenological model by Tsodyks & Markram [7] to simulate short-term synaptic depression: (1 dDk (t) = dt Dk (t)) d Uk Dk (t) (t tsp ); k (2)\n\nwhere Dk is a `depression' variable, Dk 2 [0; 1], d is the recovery time constant, Uk is a constant determining the step decrease in Dk . Using the diffusion approximation, we can get from (1) and (2) v dv (t) = dt v(t) + Se (t) + v Dk ) d\nN X\n\nADk ( k + p k\n\nk=1\n\ndDk (t) (1 = dt\n\np\n\nk\n\nk (t));\n\nUk Dk ( k +\n\nk (t)):\n\n(3)\n\nThe Fokker-Planck equation of equations (3) is @ p(t; v ; D) = @t @( @v\n2\n\nv + Kv p) v\nN X\n\nk=1\n\nX@ N (KDk p) @ Dk\nk=1 N X 2\n\nk=1\n\nX N\n\n@ 2 ( kAUk Dk p) @ v @ Dk\n\n1@ + f 2( 2 @v K v = Se +\n\n2 kA2 Dk p) +\n\nk=1\n\n@ 22 2 ( kUk Dk p)g; @ Dk (1 Dk ) d kUk Dk : (4)\n\nk=1\n\nN X\n\nv\n\nk ADk ;\n\nK Dk =\n\n\f\nwhere D = (D1 ; D2 ; :::DN ), and p(t; v ; D) = pd (t; Djv )pv (t; v );\nN Y\n\nWe assume that D1 ; D2 ; :::DN are uncorrelated, then we have pd (t; Djv ) = pk (t; Dk jv ); ~d (6)\n\nZ\n\n1 1\n\npd (t; Djv )dD = 1:\n\n(5)\n\nk=1\n\nwhere pk (t; Dk jv ) is the conditional probability density. Moreover, we can assume ~d Substituting (5) into (4), we get pd pk (t; Dk jv ) ~d pk (t; Dk ): d\n\n(7)\n\n@ pd v + Kv @ pv + pv = @( N pv p d ) @t @t @v v X X@ @ 2 pv (KDk pd ) N (AUk Dk kpv pd )+ @ Dk @ v @ Dk\nk=1 k=1\n\n1 @2 f ( 2 @ v2\n\nk=1\n\nIntegrating Eqation (8) over D, we get v where ~ Kv = mk = Z Kv pd dD =Se +\nN X\n\nN X\n\n2 kA2 Dk pv pd ) +\n\nk=1\n\nN X @2 22 2 ( kUk Dk pv pd )g: @ Dk\n\n(8)\n\n@ pv (t; v ) = @t\n\nQv @ 2 pv (t; v ) ~ @ ( v + Kv )pv (t; v ) + ; @v 2 @ v2\nN X\n\n(9)\n\nv\n\nk Amk ; Qv =\n\nv\n\n2 kA\n\nk; (10)\n\nand pk (t; Dk ) satis es the following equation Fokker-Planck equation d @ pk d = @t 1 @2 22 @ (KDk pk ) + kpk ): d d 2 (U D @ Dk 2 @ Dk k k 1 1 + U k)mk + ; d d 2 2mk 2 ( + (2U U ) k) k + : d d (\nv\n\nZ\n\nk=1\n\nDk pk (t; Dk )dDk ; d\n\nk=\n\nZ\n\nk=1 2 Dk pk (t; Dk )dDk ; d\n\n(11)\n\nFrom (10) and (11), we can get dmk = dt dk = dt Let Jv (t; v ) = ( ~ v + Kv )pv (t; v ) v r(t) = Jv (t; 1); Q 2v @ pv (t; v ) ; @v (13)\n\n(12)\n\nwhere Jv (t; v ) is the probability ux of pv , r(t) is the re rate. The boundary conditions of equation (9) are Z1 pv (t; 1) = 0; pv (t; v )dv = 1; r(t) = Jv (t; 0): (14)\n0\n\n\f\n2.2\n\nStationary Solution and Response Analysis\n\nWhen the system is in the stationary states, @ pv =@ t = 0; dmk =dt = 0; d k=dt = 0; pv (t; v ) = p0 (v ); r(t) = r0 ; mk (t) = m0 ; k(t) = 0 and k(t) = 0 . are timev k k k independent. From (9), (12), (13) and (14), we get Z1 0 2 vr0 v Kv )2 ~0 (v Kv )2 0 ~0 p0 (v ) = exp[ ( ] exp[ ]dv ; 0 v 1; v 0 0 0 Qv Qv Qv v 11 0 0 1 Kv p ~0 0 ~ K Bp C Q r0 = @ v Z K 0 v exp(u2 )[erf ( p v ) + erf (u)]duA ; ~v Q0 p v\nQ0 v\n\n~0 K v = Se +\n\nk=1\n\n1 m0 = k 1 + Uk d\n\nN X\n\nvA\n0; k\n\n00 k mk ;\n\nQ0 = v 0= k\n\nk=1\n\nSometimes, we are more interested in the instantaneous response to time-dependence random uctuation inputs. The inputs take the form: 1 k = k (1 + \"k k (t)); 0 where \"k 1. Then mk and k have the forms, i.e., mk = m0 (1 + \"k m1 (t) + O(\"2 )); k k k k = 0 (1 + \"k 1 (t) + O(\"2 )); k k k ~ and Kv and Qv are ~ K v = Se +\nN X N X\n\n2m0 k : 2 2 + d(2Uk Uk ) 0 k\n\nN X\n\nvA2 0 0 ; kk (15)\n\n(16)\n\n(17)\n\nvA\n\n00 k mk\n\n+\n\nk=1\n\nk=1 N X\n\nN X\n\n\"k vA\n\n00 k mk ( 1 k\n\n+ m1 )) + O(\"2 ); k k (18)\n\nQv =\n\nvA2 k k + 00\n\n1 \"k vA2 k k ( k + k ) + O(\"2 ): 00 1 k\n\nk=1\n\nk=1\n\nSubstituting (17) into (12), and ignoring the high order item, it yields: dm1 k = dt dk 1 = dt With the de nitions ~ ~0 Kv = Kv + Kv (t) + O( 2); ~1 Qv = Q0 + Q1 (t) + O( 2); v v pv = p0 + p1 (t) + O( 2); v r = r0 + r1 (t) + O( 2); where 1; and boundary conditions of p1 Z p1 (t; 1) = 0;\n1\n\n1 0k + Uk 0 )m1 Uk k 1 (t); k k d 2 2m1 2 k ( + (2Uk Uk ) 0 ) 1 + kk d d (\n\n(2Uk\n\n2 Uk ) 0 1 (t): kk\n\n(19)\n\n(20)\n\np1 (t; v )dv = 0;\n\n(21)\n\n0\n\n\f\nusing the perturbative expansion in powers of ; we can get Qv @ 2 p0 (v ) v ~0 v ; @ ( v + Kv )p0 (v ) + @v 2 @ v2 @ p1 f0 (t; v ) Q0 @ 2 p1 ~0 v @ = @ ( v + Kv )p1 + v ; @t @v 2 @ v2 @v 1 (t) @ p0 v ~1 f0 (t; v ) = Kv (t)p0 Qv ; v 2 @v 0 1 @ p1 (t; 1) (t) @ p0 (1) v r1 = Qv Qv : 2v @v 2v @v 0=\n\n(22)\n\n~1 For the oscillatory inputs Kv (t) = k (! )ej !t , Q1 (t) = q (! )ej !t , the output has the same v frequency and takes the forms p1 (t; v ) = p! (! ; v )ej !t ; @ p1 =@ t = j ! p1 . For inputs that vary on a slow enough time scale, satisfy v! l = v!; p1 = p0 + lp1 + O( l ); 2 1 1\n0 1 r1 = r1 + lr1 + O( l ): 2\n\n1; we de ne\n\n(23)\n\nUsing the perturbative expansion in powers of l; we get @ f0 (t; v ) = @v j p0 = 1 The solutions of equtions (24) are Z1 0 v Kv )2 ~0 2 (v Kv )2 0 ~0 n exp[ ( ] ]dv ; ( vr1 Fn ) exp[ Q0 Q0 Q0 v v v v Z Z1 0 2r0 1 v Kv )2 ~0 (v Kv )2 0 ~0 n r1 = 0 ] ]dv dv ; exp[ ( Fn exp[ 0 0 Qv 0 Qv Qv v Zv 0 0 F0 = f0 (t; v ); F1 = j p0 (v )dv ; n = 0; 1. 1 pn = 1\n0 0 ~0 1 Q @ ( v + Kv )p0 + v @v 2 Q0 ~0 @ ( v + Kv )p1 + v 1 @v 2\n\n@ 2 p0 1 ; @ v2 @ 2 p1 1 : @ v2\n\n(24)\n\n(25)\n\nIn general, Q1 (t) v\n\nKv (t), then we have ~1 F0 = f0 (t; v ) Kv (t)p0 : ~1 v (26)\n\nFrom (23), (25) and (26), we can get Z1 Z1 0 r0 ~ 1 v Kv )2 ~0 (v Kv )2 0 ~0 r1 2 0 Kv (t) exp[ ( ] p0 exp[ ]dv dv + j ! v v 0 0 Qv Qv Qv 0 v Z Z 1 Z v0 0 00 00 v Kv )2 ~0 (v Kv )2 0 ~0 2r0 1 exp[ ( ] [ p0 (v )dv ] exp[ ]dv dv : 1 0 0 0 Qv 0 Qv Qv v 0 In the limit of high frequency inputs, i.e. 1= v! h= 1; with the de nitions\n\n(27)\n\n1 ; v! p1 = p0 + hp1 + O( 2 ); h h h\n\n(28)\n\n\f\nwe obtain p0 = 0; h @ f0 (t; v ) ; @v 1 (t) @ p0 (1) Q0 @ 2 f0 (t; 1) v r1 = Qv + O( 2 ) jh v h 2v @v 2v @ v2 1 0 20 1 (t) @ 3 p0 Q ~ 1 @ pv (1) (t) v Qv ) Qv r0 j h v (Kv (t) Q0 2v @ v2 2 @ v3 ( 1 1 ~1 Q1 (t)r0 j h Kv (t)r0 (t) = v0 2 1 Kv ) ~0 Qv ~1 Qv Q0 Kv (t)Q0 v v p1 = j h Kv (t), we have ~1 r1\n1 (t)r0 Qv 0 Qv\n\nKv ~0\n\nQ0 v\n\n:\n\n(29)\n\nWhen Q1 (t) v\n\n2\n\n~1 j Kv (t)r0 (1 v!Q0 v\n\nKv )(1 ~0\n\n1 (t) Qv ); 1 (t)Q0 ~ Kv v\n\n(30)\n\n3\n\nDiscussion\n\n~0 In equation (15), Kv re ects the average intensity of background inputs and Q0 re ects the v intensity of background noise. When 1 dUk 0 , we have k ~0 Kv Q0 v Se + X N\nN X vA ; dUk\n\nk=1\n\nk=1\n\nvA2 dUk (1 + dUk 0 (1 k\n\nUk =2))\n\n:\n\n(31)\n\n~0 From (31), we can know the change of background inputs 0 has little in uence on Kv k 0 which is dominated by parameter vA= d Uk , but more in uence on Qv which decreases with 0 increasing. k In the low input frequency regime, from (27), we can know that the input frequency ! increasing will result in the response amplitude and the phase delay increasing. However, in the high input frequency limit regime, from (30), we can know the input frequency ! increasing will result in the response amplitude and the phase delay decreasing. Moreover, from (27) and (30), we know the stationary background re rate r0 play an important part in response to changes in uctuation outputs. The instantaneous response r1 increases monotonically with background re rate r0 :But rhe background re rate r0 is a function t r ~1 of the background noise Q0 : In equation (27), 1 =Kv e ects the response amplitude, v\n\nand inrequationa(30), r0 =Q0 re ects the response amplitude. As Figure 1 (A) and (B) show v ~1 ~0 that 1 =Kv nd r0 =Q0 changes with variables Q0 and Kv respectively. We can know, v v ~0 ~0 for the subthreshold regime (Kv < 1), they increase monotonically with Q0 when Kv is a v 0 ~ constant. However, for the suprathreshold regime (Kv > 1), they decrease monotonically 0 0 ~ v is a constant. When inputs remain, if the instantaneous response ampliwith Qv when K tude increases, then we can take for the role of neurons are more like coincidence detection than temporal integration. And from this viewpoint, it suggests that the background inputs play an important role in information processing and act as a switch between temporal integration and coincidence detection. In equation (16), if the inputs take the oscillatory form, 1 (t) = ej !t ; according to (19), k\n\n\f\n( r ~1 e0 Figure 1: Response amplitude versus Q0 and Kv . (A) 1 =Kv for equation (27)) v ~ ~ changes with Q0 and K 0 . (B) r0 =Q0 (for equation (30)) changes with Q0 and K 0 .\nv v v v v\n\nwe get m1 = k Uk 0 ej (!t m) k q d ; ( d!)2 + (1 + dUk 0 )2 k (32)\n\nq d! where m =arctg( 1+ dUk 0 ) is the phase delay, dUk 0 = ( d!)2 + (1 + dUk 0 )2 is k k k the amplitude. The minus shows it is a `depression' response amplitude. The phase delay increases with the input frequency ! and decreases with the background input 0 . The k `depression' response amplitude decrease with the input frequency ! and increase with the background input 0 . The equations (15) (18), (12), (19), (27), (30) and (32) show us a k point of view that the synapses can be regarded as a time-dependent external eld which impacts on the neuronal population through the time-dependent mean and variance. We 1 assume the inputs are composed of two parts, viz. 11 (t) = 12 (t) = 2 ej !t ; then we can k k 1 1 1 1 1 get mk1 and mk2 . However, in general mk 6= mk1 + mk2 , this suggest for us that the spatial distribution of synapses and inputs is important on neural information processing. In conclusion, the role of synapses can be regarded as a spatio-temporal lter. Figure 2 is the results of simulation of a network of 2000 neurons and the analytic solution for equation (15) and equation (27) in different conditions.\n\n4\n\nSummary\n\nIn this paper, we deal with the model of the integrate-and- re neurons with synaptic current dynamics and synaptic depression. In Section 2, rst, using the membrane potential equation (1) and combining the synaptic depression equation (2), we derive the evolution equation (4) of the joint distribution density function. Then, we give an approach to cut the evolution equation of the high dimensional function down to one dimension, and get equation (9). Finally, we give the stationary solution and the response of instantaneous re rate to time-dependence random uctuation inputs. In Section 3, the analysis and discussion of the model is given and several signi cant conclusions are presented. This paper can only investigate the IF neuronal model without internal connection. We can also extend to other models, such as the non-linear IF neuronal models of sparsely connected networks of excitatory and inhibitory neurons.\n\n\f\nFigure 2: Simulation of a network of 2000 neurons (thin solid line) and the analytic solution (thick solid line) for equation (15) and equation (27), with v = 15(ms), d = 1(s), A = 0:5, Uk = 0:5, N = 30, ! = 6:28(Hz); 1 = sin(! t), \"k 0 = 10(Hz), 0 = 70(Hz) k k k (A and C) and 100(Hz) (B and D), Se = 0:5(A and B) and 0:8(C and D). The horizontal axis is time (0-2s), and the longitudinal axis is the re rate.\n\nReferences\n[1] Fourcaud N. & Brunel, N. (2005) Dynamics of the Instantaneous Firing Rate in Response to Changes in Input Statistics. Journal of Computational Neuroscience 18(3):311-321. [2] Fourcaud, N. & Brunel, N. (2002) Dynamics of the Firing Probability of Noisy Integrate-and-Fire Neurons. Neural Computation 14(9):2057-2110. [3] Gerstner, W. (2000) Population Dynamics of Spiking Neurons: Fast Transients, Asynchronous States, and Locking. Neural Computation 12(1):43-89. [4] Silberberg, G., Bethge, M., Markram, H., Pawelzik, K. & Tsodyks, M. (2004) Dynamics of Population Rate Codes in Ensembles of Neocortical Neurons. J Neurophysiol 91(2):704-709. [5] Abbott, L.F. & Regehr, W.G. (2004) Synaptic Computation. Nature 431(7010):796-803. [6] Destexhe, A. & Marder, E. (2004) Plasticity in Single Neuron and Circuit Computations. Nature 431(7010):789-795. [7] Markram, H., Wang, Y. & Tsodyks, M. (1998) Differential Signaling Via the Same Axon of Neocortical Pyramidal Neurons. Proc Natl Acad Sci USA 95(9):5323-5328.\n\n\f\n", "award": [], "sourceid": 2845, "authors": [{"given_name": "Wentao", "family_name": "Huang", "institution": null}, {"given_name": "Licheng", "family_name": "Jiao", "institution": null}, {"given_name": "Shan", "family_name": "Tan", "institution": null}, {"given_name": "Maoguo", "family_name": "Gong", "institution": null}]}