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121 nips-2013-Firing rate predictions in optimal balanced networks

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Author: David G. Barrett, Sophie Denève, Christian K. Machens

Abstract: How are ﬁring rates in a spiking network related to neural input, connectivity and network function? This is an important problem because ﬁring rates are a key measure of network activity, in both the study of neural computation and neural network dynamics. However, it is a difﬁcult problem, because the spiking mechanism of individual neurons is highly non-linear, and these individual neurons interact strongly through connectivity. We develop a new technique for calculating ﬁring rates in optimal balanced networks. These are particularly interesting networks because they provide an optimal spike-based signal representation while producing cortex-like spiking activity through a dynamic balance of excitation and inhibition. We can calculate ﬁring rates by treating balanced network dynamics as an algorithm for optimising signal representation. We identify this algorithm and then calculate ﬁring rates by ﬁnding the solution to the algorithm. Our ﬁring rate calculation relates network ﬁring rates directly to network input, connectivity and function. This allows us to explain the function and underlying mechanism of tuning curves in a variety of systems. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Firing rate predictions in optimal balanced networks Sophie Den` ve e Group for Neural Theory ´ Ecole Normale Sup´ rieure e Paris, France sophie. [sent-1, score-0.376]

2 org Abstract How are ﬁring rates in a spiking network related to neural input, connectivity and network function? [sent-11, score-0.948]

3 This is an important problem because ﬁring rates are a key measure of network activity, in both the study of neural computation and neural network dynamics. [sent-12, score-0.569]

4 However, it is a difﬁcult problem, because the spiking mechanism of individual neurons is highly non-linear, and these individual neurons interact strongly through connectivity. [sent-13, score-0.576]

5 We develop a new technique for calculating ﬁring rates in optimal balanced networks. [sent-14, score-0.404]

6 These are particularly interesting networks because they provide an optimal spike-based signal representation while producing cortex-like spiking activity through a dynamic balance of excitation and inhibition. [sent-15, score-0.629]

7 We can calculate ﬁring rates by treating balanced network dynamics as an algorithm for optimising signal representation. [sent-16, score-0.748]

8 Our ﬁring rate calculation relates network ﬁring rates directly to network input, connectivity and function. [sent-18, score-0.752]

9 A large, sometimes bewildering, diversity of ﬁring rate responses to stimuli have since been observed [2], ranging from sigmoidal-shaped tuning curves [3, 4], to bump-shaped tuning curves [5], with much diversity in between [6]. [sent-21, score-0.748]

10 What is the computational role of these ﬁring rate responses and how are ﬁring rates determined by neuron dynamics, network connectivity and neural input? [sent-22, score-0.822]

11 However, most approaches have struggled to deal with the non-linearity of neural spike-generation mechanisms and the strong interaction between neurons as mediated through network connectivity. [sent-24, score-0.383]

12 These calculations have led to important insights into how neural network connectivity and input determine ﬁring rates. [sent-29, score-0.455]

13 We develop a new technique for calculating ﬁring rates, by directly identifying the non-linear structure of tightly balanced networks. [sent-31, score-0.395]

14 Balanced network theory has come to be regarded as the standard model of cortical activity [12, 13], accounting for a large proportion of observed activity through a dynamic balance of excitation and inhibition [14]. [sent-32, score-0.577]

15 Recently, it was found that tightly balanced networks are synonymous with efﬁcient coding, in which a signal is represented optimally subject to metabolic costs [15]. [sent-33, score-0.514]

16 This observation allows us, here, to interpret balanced network activity as an optimisation algorithm. [sent-34, score-0.581]

17 We use this technique to calculate ﬁring rates in a variety of balanced network models, thereby exploring the computational role and underlying network mechanisms of monotonic ﬁring rate tuning curves, bump-shaped tuning curves and tuning curve inhomogeneity. [sent-36, score-1.669]

18 2 Optimal balanced network models We calculate ﬁring rates in a balanced network consisting of N recurrently connected leaky integrate-and-ﬁre neurons (Fig. [sent-37, score-1.165]

19 , sN ), where si (t) = k δ(t − ti ) is the spike train of neuron i with spike times ti . [sent-52, score-0.566]

20 A spike is produced k k whenever the membrane potential Vi exceeds the spiking threshold Ti of neuron i. [sent-53, score-0.746]

21 The membrane potential has the following dynamics: dVi = −λVi + dt M N Fij Ij , Ωik sk + (1) j=1 k=1 where λ is the neuron leak, Ωik is connection strength from neuron k to neuron i and Fij is the connection strength from input j to neuron i [16]. [sent-55, score-1.047]

22 When a neuron spikes, the membrane potential is reset to Ri ≡ Ti + Ωii . [sent-56, score-0.364]

23 We are interested in networks where a balance of excitation and inhibition coincides with optimal signal representation. [sent-59, score-0.363]

24 Not all choices of network connectivity and spiking thresholds will give both [12, 13], but if certain conditions are satisﬁed, this can be possible. [sent-60, score-0.595]

25 rk = (4) 0 and xj is a temporal ﬁltering of the j th input ∞ xj = 0 All the excitatory and inhibitory inputs received by neuron i are included in this summation (Eqn. [sent-64, score-0.353]

26 Now, we can use this expression to derive the conditions that connectivity must satisfy so that the network operates in an optimal balanced state. [sent-67, score-0.569]

27 In balanced networks, excitation and inhibition cancel to produce an input that is the same order of magnitude as the spiking threshold. [sent-68, score-0.664]

28 In tightly balanced networks, which we consider, this cancellation is so precise that Vi → 0 in the large network limit (for all active neurons) [15, 17, 18]. [sent-70, score-0.554]

29 This has two implications for our choice of network connectivity and spiking thresholds. [sent-73, score-0.595]

30 Secondly, the spiking threshold of each neuron must be chosen so that each spike acts to minimise the cost function. [sent-76, score-0.637]

31 Finally, (A) (C) x x ˆ x, x ˆ (B) time (sec) Figure 1: Optimal balanced network example. [sent-79, score-0.394]

32 (A) Schematic of a balanced neural network providing an optimal spike-based representation x of a signal x. [sent-80, score-0.554]

33 (B) A tightly balanced network can ˆ produce an output x1 (blue, top panel) that closely matches the signal x1 (black, top panel). [sent-81, score-0.616]

34 Popˆ ulation spiking activity is represented here using a raster plot (middle panel), where each spike is represented with a dot. [sent-82, score-0.455]

35 For a randomly chosen neuron (red, middle panel), we plot the total excitatory input (green, bottom panel) and the total inhibitory input (red, bottom panel). [sent-83, score-0.456]

36 The sum of excitation and inhibition (black, bottom panel) ﬂuctuates about the spiking threshold (thin black line, bottom panel) indicating that this network is tightly balanced. [sent-84, score-0.803]

37 A spike is produced whenever this sum exceeds the spiking threshold. [sent-85, score-0.382]

38 (C) Firing rate tuning curves are measured during simulations of our balanced network. [sent-86, score-0.632]

39 Therefore, the spiking threshold for each neuron must be set to Tk ≡ −Ωkk /2, though this condition can be relaxed considerably if our loss function has an additional linear cost term1 . [sent-90, score-0.452]

40 We are interested in networks that are both tightly balanced and optimal. [sent-92, score-0.39]

41 This is an important result, because it relates balanced network dynamics to a neural computation. [sent-95, score-0.486]

42 Speciﬁcally, it allows us to interpret the spiking activity of our tightly balanced network as an algorithm that optimises a loss function (Eqn. [sent-96, score-0.901]

43 (8) The second term of equation 7 is a metabolic cost term that penalises neurons for spiking excessively, ˆ and the ﬁrst term quantiﬁes the difference between the signal value x and a linear read-out, x, where ˆ x is computed using the linear decoder FT (Eqn. [sent-103, score-0.572]

44 Therefore, a network with this connectivity ˆ produces spike trains that optimise equation 7, thereby producing an output x that is close to the signal value x. [sent-105, score-0.661]

45 We ﬁnd that our network produces spike trains (Fig. [sent-108, score-0.353]

46 We measure ﬁring rate tuning curves using a ﬁxed value of x2 while varying x1 . [sent-114, score-0.416]

47 We use this signal because it can produce interesting, non-linear tuning curves (Fig. [sent-115, score-0.426]

48 In the next section, we will attempt to understand this tuning curve non-linearity by calculating ﬁring rates analytically. [sent-117, score-0.456]

49 3 Firing rate analysis with quadratic programming Our goal is to calculate the ﬁring rates f of all the neurons in these tightly balanced network models as a function of the network input, the recurrent network connectivity Ω, and the feedforward connectivity F. [sent-118, score-1.767]

50 On the surface, this may seem to be a difﬁcult problem, because individual neurons have complicated non-linear integrate-and-ﬁre dynamics and they interact strongly through network connectivity. [sent-119, score-0.399]

51 Then, we ﬁnd that the optimal spiking thresholds for this network are given by Ti ≡ (−Ωii + bi )/2 ≥ −Ωii /2. [sent-127, score-0.42]

52 We can now calculate ﬁring rates using this relationship and by exploiting the algorithmic nature of tightly balanced networks. [sent-133, score-0.566]

53 Therefore, the ﬁring rates of our network are those that minimise E(f /λ), under the constraint that ﬁring rates must be positive: {fi } = arg min E(f /λ) . [sent-136, score-0.497]

54 When both neurons are active, we can solve equation 10 exactly, to see that ﬁring rates are related to network connectivity according to f = −λΩ−1 Fx. [sent-146, score-0.696]

55 When one of the neurons becomes silent, the other neuron must compensate by adjusting its ﬁring rate slope. [sent-147, score-0.461]

56 For example, when neuron 1 becomes silent, we have f1 = 0 and the ﬁring rate of neuron 2 increases to f2 = λF2 x/(F2 FT + βI), where F2 denotes the second row of F. [sent-148, score-0.504]

57 Predicted ﬁring rates closely match measured ﬁring rates for both neurons, and for all signal values (right). [sent-153, score-0.368]

58 We also measure the ﬁring rate trajectory (right panel) as the network evolves towards the minimum of the cost function E(x1 = 1) (blue cross, right panel), where neuron 2 is silent. [sent-157, score-0.472]

59 In general, we can interpret tuning curve shape to be the solution of a quadratic programming problem, which can be written as a piece-wise linear function f = M (x) · x, where M(x) is a matrix whose entries depend on the region of signal space occupied by x. [sent-163, score-0.476]

60 For example, in the two-neuron system that we just discussed, the signal space is partitioned into three regions: one region where neuron 1 is active and where neuron 2 is silent, a second region where both neurons are active and a third region where neuron 1 is silent and neuron 2 is active (Fig. [sent-164, score-1.342]

61 The boundaries of these regions occur at points in signal space where an active neuron becomes silent (or where a silent neuron becomes active). [sent-167, score-0.786]

62 We can also use quadratic programming to describe the spiking dynamics underlying these nonlinear networks. [sent-169, score-0.376]

63 The step-size is λ because when neuron i spikes, ri → ri + 1, according to equation 3, and therefore, fi → fi +λ, according to equation 9. [sent-173, score-0.41]

64 Eventually, when the ﬁring rate has decayed too far from the optimal solution, another spike is ﬁred and the network moves closer to the optimum. [sent-178, score-0.402]

65 In this way, spiking dynamics can be interpreted as a quadratic programming algorithm. [sent-179, score-0.376]

66 4 Analysing tuning curve shape with quadratic programming Now that we have a framework for relating ﬁring rates to network connectivity and input, we can explore the computational function of tuning curve shapes and the network mechanisms that generate these tuning curves. [sent-183, score-1.484]

67 We will investigate systems that have monotonic tuning curves and systems that have bump-shaped tuning curves, which together constitute a large proportion of ﬁring rate observations [2, 3, 4, 5]. [sent-184, score-0.653]

68 We begin by considering a system of monotonic tuning curves, similar to the examples that we have considered already where recurrent connectivity is given by Ω = −FFT − βI. [sent-185, score-0.469]

69 In these systems, the recurrent connectivity and hence the tuning curve shape is largely determined by the form of the feedforward matrix F. [sent-186, score-0.475]

70 This matrix also determines the contribution of tuning curves to computational function, through its role as a linear decoder for signal representation (Eqn. [sent-187, score-0.454]

71 This system produces monotonically increasing and decreasing tuning curves (Fig. [sent-191, score-0.357]

72 3, blue tuning curves), and neurons with negative F values have negative ﬁring rate slopes (Fig. [sent-194, score-0.519]

73 If the values of F are regularly spaced, then the tuning curves of individual neurons are regularly spaced, and, if we manipulate this regularity by adding some random noise to the connectivity, we obtain inhomogeneous and highly irregular tuning curves (Fig. [sent-196, score-0.99]

74 This inhomogeneous monotonic tuning is reminiscent of tuning in many neural systems, including the oculomotor system [4]. [sent-199, score-0.606]

75 The oculomotor system represents eye position, using neurons with negative slopes to represent left side eye positions and neurons with positive slopes to represent right side eye positions. [sent-200, score-0.714]

76 (A) Each dot represents the contribution of a neuron to a signal representation (when the ﬁring rate is 10 × 16 Hz) (1st column). [sent-202, score-0.416]

77 We simulate a network of neurons and measure ﬁring rates (2nd column). [sent-204, score-0.482]

78 These measurements closely match our algorithmically predicted ﬁring rates (3rd column), where each point in the 4th column represents the ﬁring rate of an individual neuron for a given stimulus. [sent-205, score-0.485]

79 The representation error (bottom panels, column 2 and column 3) is similar to the network without connectivity noise. [sent-207, score-0.437]

80 Each dot represents the contribution of a neuron to a signal representation (when the ﬁring rate is 20 × 16 Hz) (1st column). [sent-209, score-0.416]

81 This signal produces bump-shaped tuning curves (2nd column), which we can also predict accurately (3rd and 4th column). [sent-210, score-0.426]

82 (A) (B) leak membrane potential noise ⌘ Figure 4: Performance of quadratic programming in ﬁring rate prediction. [sent-211, score-0.357]

83 Now, we can use the relationship that we have developed between tuning curves and computational function to interpret oculomotor tuning as an attempt to represent eye positions optimally. [sent-218, score-0.719]

84 Bump-shaped tuning curves can be produced by networks representing circular variables x1 = cos θ, x2 = sin θ, where θ is the orientation of the signal (Fig. [sent-219, score-0.472]

85 As before, the tuning curves of individual neurons are regularly spaced if the values of F are regularly spaced. [sent-221, score-0.634]

86 If we add some noise to the connectivity F, the tuning curves become inhomogeneous and highly irregular. [sent-222, score-0.562]

87 The success of our algorithmic approach in calculating ﬁring rates depends on the success of spiking networks in algorithmically optimising a cost function. [sent-226, score-0.532]

88 The resolution of this spiking algorithm is determined by the leak λ and membrane potential noise. [sent-227, score-0.435]

89 5 Discussion and Conclusions We have developed a new algorithmic technique for calculating ﬁring rates in tightly balanced networks. [sent-233, score-0.532]

90 Identifying such relationships is a long-standing problem in systems neuroscience, largely because the mathematical language that we use to describe information representation is very different to the language that we use to describe neural network spiking statistics. [sent-236, score-0.486]

91 For tightly balanced networks, we have essentially solved this problem, by matching the ﬁring rate statistics of neural activity to the structure of neural signal representation. [sent-237, score-0.671]

92 Previous studies have also interpreted ﬁring rates to be the result of a constrained optimisation problem [21], but for a population coding model, not for a network of spiking neurons. [sent-239, score-0.691]

93 In a more recent study, a spiking network was used to solve an optimisation problem, although this network required positive and negative spikes, which is difﬁcult to reconcile with biological spiking [22]. [sent-240, score-0.92]

94 The ﬁring rate tuning curves that we calculate have allowed us to investigate poorly understood features of experimentally recorded tuning curves. [sent-241, score-0.655]

95 In particular, we have been able to evaluate the impact of tuning curve inhomogeneity on neural computation. [sent-242, score-0.381]

96 We ﬁnd that tuning curve inhomogeneity is not necessarily noise because it does not necessarily harm signal representation. [sent-244, score-0.437]

97 Therefore, we propose that tuning curves are inhomogeneous simply because they can be. [sent-245, score-0.387]

98 3 Membrane potential noise can be included in our network model by adding a Wiener process noise term to our membrane potential equation (Eqn. [sent-249, score-0.401]

99 (2006) Neocortical network activity in vivo is generated through a dynamic balance of excitation and inhibition. [sent-330, score-0.397]

100 (2012) Predictive coding of dynamical variables in balanced spiking networks. [sent-345, score-0.486]

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