nips nips2001 nips2001-72 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Julian Eggert, Berthold Bäuml
Abstract: Mesoscopical, mathematical descriptions of dynamics of populations of spiking neurons are getting increasingly important for the understanding of large-scale processes in the brain using simulations. In our previous work, integral equation formulations for population dynamics have been derived for a special type of spiking neurons. For Integrate- and- Fire type neurons , these formulations were only approximately correct. Here, we derive a mathematically compact, exact population dynamics formulation for Integrate- and- Fire type neurons. It can be shown quantitatively in simulations that the numerical correspondence with microscopically modeled neuronal populations is excellent. 1 Introduction and motivation The goal of the population dynamics approach is to model the time course of the collective activity of entire populations of functionally and dynamically similar neurons in a compact way, using a higher descriptionallevel than that of single neurons and spikes. The usual observable at the level of neuronal populations is the populationaveraged instantaneous firing rate A(t), with A(t)6.t being the number of neurons in the population that release a spike in an interval [t, t+6.t). Population dynamics are formulated in such a way, that they match quantitatively the time course of a given A(t), either gained experimentally or by microscopical, detailed simulation. At least three main reasons can be formulated which underline the importance of the population dynamics approach for computational neuroscience. First, it enables the simulation of extensive networks involving a massive number of neurons and connections, which is typically the case when dealing with biologically realistic functional models that go beyond the single neuron level. Second, it increases the analytical understanding of large-scale neuronal dynamics , opening the way towards better control and predictive capabilities when dealing with large networks. Third, it enables a systematic embedding of the numerous neuronal models operating at different descriptional scales into a generalized theoretic framework, explaining the relationships, dependencies and derivations of the respective models. Early efforts on population dynamics approaches date back as early as 1972, to the work of Wilson and Cowan [8] and Knight [4], which laid the basis for all current population-averaged graded-response models (see e.g. [6] for modeling work using these models). More recently, population-based approaches for spiking neurons were developed, mainly by Gerstner [3, 2] and Knight [5]. In our own previous work [1], we have developed a theoretical framework which enables to systematize and simulate a wide range of models for population-based dynamics. It was shown that the equations of the framework produce results that agree quantitatively well with detailed simulations using spiking neurons, so that they can be used for realistic simulations involving networks with large numbers of spiking neurons. Nevertheless, for neuronal populations composed of Integrate-and-Fire (I&F;) neurons, this framework was only correct in an approximation. In this paper, we derive the exact population dynamics formulation for I&F; neurons. This is achieved by reducing the I&F; population dynamics to a point process and by taking advantage of the particular properties of I&F; neurons. 2 2.1 Background: Integrate-and-Fire dynamics Differential form We start with the standard Integrate- and- Fire (I&F;) model in form of the wellknown differential equation [7] (1) which describes the dynamics of the membrane potential Vi of a neuron i that is modeled as a single compartment with RC circuit characteristics. The membrane relaxation time is in this case T = RC with R being the membrane resistance and C the membrane capacitance. The resting potential v R est is the stationary potential that is approached in the no-input case. The input arriving from other neurons is described in form of a current ji. In addition to eq. (1), which describes the integrate part of the I&F; model, the neuronal dynamics are completed by a nonlinear step. Every time the membrane potential Vi reaches a fixed threshold () from below, Vi is lowered by a fixed amount Ll > 0, and from the new value of the membrane potential integration according to eq. (1) starts again. if Vi(t) = () (from below) . (2) At the same time, it is said that the release of a spike occurred (i.e., the neuron fired), and the time ti = t of this singular event is stored. Here ti indicates the time of the most recent spike. Storing all the last firing times , we gain the sequence of spikes {t{} (spike ordering index j, neuronal index i). 2.2 Integral form Now we look at the single neuron in a neuronal compound. We assume that the input current contribution ji from presynaptic spiking neurons can be described using the presynaptic spike times tf, a response-function ~ and a connection weight W¡ . ',J ji(t) = Wi ,j ~(t - tf) (3) l: l: j f Integrating the I&F; equation (1) beginning at the last spiking time tT, which determines the initial condition by Vi(ti) = vi(ti - 0) - 6., where vi(ti - 0) is the membrane potential just before the neuron spikes, we get 1 Vi(t) = v Rest + fj(t - t:) + l: Wi ,j l: a(t - t:; t - tf) , j - Vi(t:)) e- S / T (4) f with the refractory function fj(s) = - (v Rest (5) and the alpha-function r ds
Reference: text
sentIndex sentText sentNum sentScore
1 Exact differential equation population dynamics for Integrate-and-Fire neurons Julian Eggert * HONDA R&D; Europe (Deutschland) GmbH Future Technology Research Carl-Legien-StraBe 30 63073 Offenbach/Main, Germany julian. [sent-1, score-1.053]
2 de Abstract Mesoscopical, mathematical descriptions of dynamics of populations of spiking neurons are getting increasingly important for the understanding of large-scale processes in the brain using simulations. [sent-9, score-1.037]
3 In our previous work, integral equation formulations for population dynamics have been derived for a special type of spiking neurons. [sent-10, score-0.888]
4 For Integrate- and- Fire type neurons , these formulations were only approximately correct. [sent-11, score-0.333]
5 Here, we derive a mathematically compact, exact population dynamics formulation for Integrate- and- Fire type neurons. [sent-12, score-0.678]
6 It can be shown quantitatively in simulations that the numerical correspondence with microscopically modeled neuronal populations is excellent. [sent-13, score-0.51]
7 1 Introduction and motivation The goal of the population dynamics approach is to model the time course of the collective activity of entire populations of functionally and dynamically similar neurons in a compact way, using a higher descriptionallevel than that of single neurons and spikes. [sent-14, score-1.687]
8 The usual observable at the level of neuronal populations is the populationaveraged instantaneous firing rate A(t), with A(t)6. [sent-15, score-0.456]
9 t being the number of neurons in the population that release a spike in an interval [t, t+6. [sent-16, score-1.093]
10 Population dynamics are formulated in such a way, that they match quantitatively the time course of a given A(t), either gained experimentally or by microscopical, detailed simulation. [sent-18, score-0.418]
11 At least three main reasons can be formulated which underline the importance of the population dynamics approach for computational neuroscience. [sent-19, score-0.645]
12 First, it enables the simulation of extensive networks involving a massive number of neurons and connections, which is typically the case when dealing with biologically realistic functional models that go beyond the single neuron level. [sent-20, score-0.572]
13 Second, it increases the analytical understanding of large-scale neuronal dynamics , opening the way towards better control and predictive capabilities when dealing with large networks. [sent-21, score-0.468]
14 Third, it enables a systematic embedding of the numerous neuronal models operating at different descriptional scales into a generalized theoretic framework, explaining the relationships, dependencies and derivations of the respective models. [sent-22, score-0.194]
15 Early efforts on population dynamics approaches date back as early as 1972, to the work of Wilson and Cowan [8] and Knight [4], which laid the basis for all current population-averaged graded-response models (see e. [sent-23, score-0.621]
16 More recently, population-based approaches for spiking neurons were developed, mainly by Gerstner [3, 2] and Knight [5]. [sent-26, score-0.504]
17 It was shown that the equations of the framework produce results that agree quantitatively well with detailed simulations using spiking neurons, so that they can be used for realistic simulations involving networks with large numbers of spiking neurons. [sent-28, score-0.617]
18 Nevertheless, for neuronal populations composed of Integrate-and-Fire (I&F;) neurons, this framework was only correct in an approximation. [sent-29, score-0.402]
19 In this paper, we derive the exact population dynamics formulation for I&F; neurons. [sent-30, score-0.678]
20 This is achieved by reducing the I&F; population dynamics to a point process and by taking advantage of the particular properties of I&F; neurons. [sent-31, score-0.621]
21 The membrane relaxation time is in this case T = RC with R being the membrane resistance and C the membrane capacitance. [sent-34, score-1.101]
22 The resting potential v R est is the stationary potential that is approached in the no-input case. [sent-35, score-0.276]
23 The input arriving from other neurons is described in form of a current ji. [sent-36, score-0.34]
24 (1), which describes the integrate part of the I&F; model, the neuronal dynamics are completed by a nonlinear step. [sent-38, score-0.445]
25 Every time the membrane potential Vi reaches a fixed threshold () from below, Vi is lowered by a fixed amount Ll > 0, and from the new value of the membrane potential integration according to eq. [sent-39, score-1.136]
26 (2) At the same time, it is said that the release of a spike occurred (i. [sent-42, score-0.435]
27 , the neuron fired), and the time ti = t of this singular event is stored. [sent-44, score-0.4]
28 Here ti indicates the time of the most recent spike. [sent-45, score-0.254]
29 Storing all the last firing times , we gain the sequence of spikes {t{} (spike ordering index j, neuronal index i). [sent-46, score-0.411]
30 2 Integral form Now we look at the single neuron in a neuronal compound. [sent-48, score-0.343]
31 We assume that the input current contribution ji from presynaptic spiking neurons can be described using the presynaptic spike times tf, a response-function ~ and a connection weight W¡ . [sent-49, score-0.958]
32 ',J ji(t) = Wi ,j ~(t - tf) (3) l: l: j f Integrating the I&F; equation (1) beginning at the last spiking time tT, which determines the initial condition by Vi(ti) = vi(ti - 0) - 6. [sent-50, score-0.315]
33 (6) If we start the integration at the time ti* of the spike before the last spike, then for ti* :::; t < ti the membrane potential is given by an expression like eq. [sent-52, score-1.106]
34 Especially we can now express v( ti - 0) and therefore the initial condition for an integration starting at tT in terms of ti* and v(ti* - 0). [sent-54, score-0.251]
35 n(t) to the membrane potential indicate refractory and synaptic components to the neuron i, respectively, as normally used in the Spike- Response- Model (SRM) notation [2]. [sent-59, score-0.918]
36 Both equations (4) and (7) formulate the neuronal dynamics using a refractory component, which depends on the own spike releases of a neuron, and a synaptic component, which comprises the integrated input contribution to the membrane potential by arrival of spikes from other neurons 2. [sent-60, score-2.088]
37 The synaptic component is based on the alpha-function characteristic of isolated arriving spikes, with an increase of the membrane potential after spike arrival and a subsequent exponential decrease. [sent-61, score-0.953]
38 2S0 the I&F; model can be formulated as a special case of the Spike- Response- Model, which defines the neuronal dynamics in the integral formulation. [sent-64, score-0.499]
39 (4), the refractory component depends only on the time elapsed since the last spike (thus reflecting a renewal property, sometimes also called a short term memory for refractory properties), whereas in eq. [sent-69, score-0.917]
40 The simpler form of the refractory contribution in eq. [sent-71, score-0.238]
41 (4) is achieved at the cost of an alpha-function that now depends on the time t - ti elapsed since the last own spike in addition to the times t - tf elapsed since the release of spikes at the presynaptic neurons j that provide input to i. [sent-72, score-1.466]
42 (7) , we have a more complex refractory contribution, but an alpha-function that does not depend on the last own spike time any more. [sent-74, score-0.575]
43 3 Probabilistic spike release Probabilistic firing is introduced into the I&F; model eq. [sent-76, score-0.466]
44 The spike release of each neuron is controlled by a hazard function >. [sent-79, score-0.671]
45 that a neuron with membrane potential v spikes in [t , t + dt) (9) When a neuron spikes, we proceed as usual: The membrane potential is reset by a fixed amount 6. [sent-82, score-1.463]
46 1 Density description Descriptions of neuronal populations usually assume a neuronal density function p(t; X) which depends on the state variables X of the neurons. [sent-85, score-0.779]
47 The density quantifies the likelihood that a neuron picked out of the population will be found in the vicinity of the point X in state space, p(t; X) dX = Portion of neurons at time t with state in [X, X + dX) (10) If we know p(t; X) , the population activity A(t) can be easily calculated. [sent-86, score-1.488]
48 Using the hazard function >'(t; X), the instantaneous population activity (spikes per time) can be calculated by computing the spike release averaged over the population, A(t) = J dX >. [sent-87, score-1.004]
49 (t; X) p(t; X) (11) The population dynamics is then given by the time course of the neuronal density function p(t; X), which changes because each neuron evolves according to its own internal dynamics, e. [sent-88, score-1.143]
50 after a spike release and the subsequent reset of the membrane potential. [sent-90, score-0.808]
51 The main challenge for the formulation of a population dynamics resides in selecting a low-dimensional state space [for an easy calculation of A(t)] and a suitable form for gtp(t; X). [sent-91, score-0.729]
52 As an example, for the population dynamics for I&F; neurons it would be straightforward to use the membrane potential v from eq. [sent-92, score-1.4]
53 But this leads to a complicated density dynamics, because the dynamics for v(t) consist of a continuous (differential equation (1)) and a discrete part (spike generation). [sent-94, score-0.397]
54 2 Exact population dynamics for I&F; neurons Which is the best state space for a population dynamics of I&F; neurons? [sent-97, score-1.608]
55 For the formulation of a population dynamics, it is usually assumed that the synaptic contributions to the membrane potential are identical for all neurons. [sent-98, score-1.011]
56 This is the case if we group all neurons of the same dynamical type and with identical connectivity patterns into one population. [sent-99, score-0.333]
57 (4), we see that , since o:(s, s') depends on s = t - ti and therefore on the own last spike time, the synaptic contribution to the membrane potential differs according to the state of the neuron. [sent-102, score-1.287]
58 Here, we see that for identical connectivity patterns Wi,j, the synaptic contributions are the same for all neurons, because 0:(00, s') does not depend on the own spike time any more. [sent-105, score-0.522]
59 We see that, for a fixed synaptic contribution, the membrane potential Vi is fully determined by the set of the own past spiking times {tf}. [sent-108, score-0.85]
60 But this would mean an infinite-dimensional density for the state description of a population, and, accordingly, a computationally overly expensive calculation of the population activity A(t) according to eq. [sent-109, score-0.658]
61 (8), the single spike refractory contributions 'TJ(s) are exponential. [sent-113, score-0.552]
62 (7) (12) Now the membrane potential Vi(t) only depends on the time of the last own spike ti and the refractory contribution amplitude modulation factor at the last spike 'TJi . [sent-115, score-1.698]
63 In addition, we have to care about updating of ti and 'TJi when a neuron spikes so that we get 3 'TJi --+ 'TJi = 1 + 'TJie -(t-tn! [sent-117, score-0.489]
64 The effect of taking into account more than the most recent spike ti in the refractory component vief(t) leads to a modulation factor 'TJi greater than 1, in particular if spikes come in a rapid succession so that refractory contributions can accumulate. [sent-119, score-1.111]
65 Instead of using a modulation factor 'TJi the effect of previous spikes can also be taken into account by introducing an effective last spiking time ii. [sent-120, score-0.484]
66 Because of 'TJi ::::: 1 it holds for the effective last spiking time ii ::::: ti. [sent-122, score-0.317]
67 This means, that , while at a given time t it is allways ti :::; t, it happens that ii ::::: t, meaning the neurons behave as if they would spike in the future. [sent-123, score-0.852]
68 For the membrane potential we get now instead of eq. [sent-125, score-0.468]
69 : 0:(00; t - t;) (16) f j and for the update rule for the effective last spiking time *) * * tA - +tA = f (t 'tA ' i i i with t; follows (17) (18) Therefore we can regard the dimensionality of the state space of the I&F; dynamics as 1-dimensional in the description of eq. [sent-130, score-0.695]
70 The dynamics of the single I&F; neurons now turns out to be very simple: Calculate the membrane potential Vi(t) using eq. [sent-132, score-1.079]
71 If the membrane potential exceeds threshold, update according to eq. [sent-135, score-0.518]
72 t; Using this single neuron dynamics , we can now proceed to gain a population dynamics using a density p(t; t*). [sent-137, score-1.21]
73 By fixing t* and the synaptic contribution vsyn(t) to the membrane potential, the state of a neuron is fully determined and the hazard function can be written as ,X[vs yn(t); t*]. [sent-139, score-0.801]
74 The dynamics of the density p(t; t*) is then calculated as follows. [sent-140, score-0.404]
75 Changes of p(t; t*) occur when neurons spike and t* is updated according to eq. [sent-141, score-0.623]
76 The hazard function controls the spike release, and, therefore, the change of p(t; t*). [sent-143, score-0.377]
77 For chosen state variables, p(t; t*) decreases due to spiking of the neurons with the fixed t*, and increases because neurons with other t'* spike and get updated in just that way that after updating their state variable falls around t*. [sent-144, score-1.241]
78 (11) as follows 1 +00 A(t) = - 00 dt* ,X[vsyn(t); t*] p(t; t*) (21) Remark that the expression for the density dynamics (eq. [sent-148, score-0.375]
79 20) automatically conserves the norm of the density, so that 1 +00 - 00 dt* p( t ; t*) = const , (22) which is a necessary condition because the number of neurons participating in the dynamics must remain constant. [sent-149, score-0.585]
80 This means that all we have to store is the density p(t; i*) for past and future effective last spiking times i* 4 . [sent-151, score-0.416]
81 The activity A(t) only appears as an auxiliary variable that is calculated with the help of the neuronal density. [sent-153, score-0.28]
82 In figure 1 the simulation results for populations of of spiking neurons are shown. [sent-154, score-0.755]
83 The neurons are uncoupled and a hazard function A(V) = ~ e2,B(v-e) (23) , TO with spike rate at threshold liTO = 1. [sent-155, score-0.758]
84 (1) are: resting potential v Rest = 0, jump in membrane potential after spike release ~ = 1 and time constant T = 20ms. [sent-160, score-1.101]
85 ~,' j ~ I:, r------- vsyr'i-'_ _100 _ _ __0_ _ 0 _ _ 50_ _ 0 ----,1 15 20_ 2_ 30_ (ms) o~ I c) 100 150 200 250 300 I (ms) Figure 1: Activity A(t) of simulated populations of neurons. [sent-184, score-0.231]
86 The neurons are uncoupled and to each neuron the same synaptic field vsyn(t), ploted in c) and d), is applied. [sent-185, score-0.615]
87 a) shows the activity A(t) for a population of I&F; neurons simulated on the one hand as N = 10000 single neurons (solid line) using eq. [sent-186, score-1.075]
88 (7) and on the other hand using the density dynamics eq. [sent-187, score-0.375]
89 In b) the activity A(t) of a population ofI&F; neurons (dashed line) and a population of SRM neurons with renewal (solid line) are compared. [sent-189, score-1.45]
90 (20) reproduces the activity A( t) of a population of single I&F; neurons almost perfect, with the exception of the noise in the single neuron simulations due to the finite size effects. [sent-192, score-1.003]
91 This holds even for the peaks occuring at the steps of the applied synaptic field v syn (t), although the density dynamics is entirely based on differential equations and one would therefore not expect such an excellent match for fast changes in activity. [sent-193, score-0.68]
92 The simulations also show that there can be a big difference between I&F; and SRM neurons with renewal. [sent-195, score-0.378]
93 Because of the accumulation of the refractory effects of all former spikes in the case of I&F; neurons the activity A(t) is generaly lower than for the SRM neurons with renewal and the higher the absolute actitvity level the bigger is the difference between both. [sent-196, score-1.079]
94 5 Conclusions In this paper we derived an exact differential equation density dynamics for a population of I&F; neurons starting from the microscopical equations for a single neuron. [sent-197, score-1.283]
95 This density dynamics allows a compuationaly efficient simulation of a whole population of neurons. [sent-198, score-0.742]
96 In future work we want to simulate a network of connected neuronal populations. [sent-199, score-0.191]
97 by x) , a self-consistent system of differential equations based on the population's p(x, t; i*) and A(x, t) emerges if we constrain ourselves to neuronal populations connected synaptically according to the constraints given by the pool definition [2]. [sent-202, score-0.559]
98 In this case, two neurons i and j belong to pools x and y, if Wi,j = W(x, y). [sent-203, score-0.311]
99 Population dynamics of spiking neurons: Fast transients, asynchronous states and locking. [sent-215, score-0.467]
100 Excitatory and inhibitory interactions in localized populations of model neurons. [sent-251, score-0.231]
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simIndex simValue paperId paperTitle
same-paper 1 0.99999952 72 nips-2001-Exact differential equation population dynamics for integrate-and-fire neurons
Author: Julian Eggert, Berthold Bäuml
Abstract: Mesoscopical, mathematical descriptions of dynamics of populations of spiking neurons are getting increasingly important for the understanding of large-scale processes in the brain using simulations. In our previous work, integral equation formulations for population dynamics have been derived for a special type of spiking neurons. For Integrate- and- Fire type neurons , these formulations were only approximately correct. Here, we derive a mathematically compact, exact population dynamics formulation for Integrate- and- Fire type neurons. It can be shown quantitatively in simulations that the numerical correspondence with microscopically modeled neuronal populations is excellent. 1 Introduction and motivation The goal of the population dynamics approach is to model the time course of the collective activity of entire populations of functionally and dynamically similar neurons in a compact way, using a higher descriptionallevel than that of single neurons and spikes. The usual observable at the level of neuronal populations is the populationaveraged instantaneous firing rate A(t), with A(t)6.t being the number of neurons in the population that release a spike in an interval [t, t+6.t). Population dynamics are formulated in such a way, that they match quantitatively the time course of a given A(t), either gained experimentally or by microscopical, detailed simulation. At least three main reasons can be formulated which underline the importance of the population dynamics approach for computational neuroscience. First, it enables the simulation of extensive networks involving a massive number of neurons and connections, which is typically the case when dealing with biologically realistic functional models that go beyond the single neuron level. Second, it increases the analytical understanding of large-scale neuronal dynamics , opening the way towards better control and predictive capabilities when dealing with large networks. Third, it enables a systematic embedding of the numerous neuronal models operating at different descriptional scales into a generalized theoretic framework, explaining the relationships, dependencies and derivations of the respective models. Early efforts on population dynamics approaches date back as early as 1972, to the work of Wilson and Cowan [8] and Knight [4], which laid the basis for all current population-averaged graded-response models (see e.g. [6] for modeling work using these models). More recently, population-based approaches for spiking neurons were developed, mainly by Gerstner [3, 2] and Knight [5]. In our own previous work [1], we have developed a theoretical framework which enables to systematize and simulate a wide range of models for population-based dynamics. It was shown that the equations of the framework produce results that agree quantitatively well with detailed simulations using spiking neurons, so that they can be used for realistic simulations involving networks with large numbers of spiking neurons. Nevertheless, for neuronal populations composed of Integrate-and-Fire (I&F;) neurons, this framework was only correct in an approximation. In this paper, we derive the exact population dynamics formulation for I&F; neurons. This is achieved by reducing the I&F; population dynamics to a point process and by taking advantage of the particular properties of I&F; neurons. 2 2.1 Background: Integrate-and-Fire dynamics Differential form We start with the standard Integrate- and- Fire (I&F;) model in form of the wellknown differential equation [7] (1) which describes the dynamics of the membrane potential Vi of a neuron i that is modeled as a single compartment with RC circuit characteristics. The membrane relaxation time is in this case T = RC with R being the membrane resistance and C the membrane capacitance. The resting potential v R est is the stationary potential that is approached in the no-input case. The input arriving from other neurons is described in form of a current ji. In addition to eq. (1), which describes the integrate part of the I&F; model, the neuronal dynamics are completed by a nonlinear step. Every time the membrane potential Vi reaches a fixed threshold () from below, Vi is lowered by a fixed amount Ll > 0, and from the new value of the membrane potential integration according to eq. (1) starts again. if Vi(t) = () (from below) . (2) At the same time, it is said that the release of a spike occurred (i.e., the neuron fired), and the time ti = t of this singular event is stored. Here ti indicates the time of the most recent spike. Storing all the last firing times , we gain the sequence of spikes {t{} (spike ordering index j, neuronal index i). 2.2 Integral form Now we look at the single neuron in a neuronal compound. We assume that the input current contribution ji from presynaptic spiking neurons can be described using the presynaptic spike times tf, a response-function ~ and a connection weight W¡ . ',J ji(t) = Wi ,j ~(t - tf) (3) l: l: j f Integrating the I&F; equation (1) beginning at the last spiking time tT, which determines the initial condition by Vi(ti) = vi(ti - 0) - 6., where vi(ti - 0) is the membrane potential just before the neuron spikes, we get 1 Vi(t) = v Rest + fj(t - t:) + l: Wi ,j l: a(t - t:; t - tf) , j - Vi(t:)) e- S / T (4) f with the refractory function fj(s) = - (v Rest (5) and the alpha-function r ds
2 0.27441451 141 nips-2001-Orientation-Selective aVLSI Spiking Neurons
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