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99 nips-2009-Functional network reorganization in motor cortex can be explained by reward-modulated Hebbian learning

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Author: Steven Chase, Andrew Schwartz, Wolfgang Maass, Robert A. Legenstein

Abstract: The control of neuroprosthetic devices from the activity of motor cortex neurons beneﬁts from learning effects where the function of these neurons is adapted to the control task. It was recently shown that tuning properties of neurons in monkey motor cortex are adapted selectively in order to compensate for an erroneous interpretation of their activity. In particular, it was shown that the tuning curves of those neurons whose preferred directions had been misinterpreted changed more than those of other neurons. In this article, we show that the experimentally observed self-tuning properties of the system can be explained on the basis of a simple learning rule. This learning rule utilizes neuronal noise for exploration and performs Hebbian weight updates that are modulated by a global reward signal. In contrast to most previously proposed reward-modulated Hebbian learning rules, this rule does not require extraneous knowledge about what is noise and what is signal. The learning rule is able to optimize the performance of the model system within biologically realistic periods of time and under high noise levels. When the neuronal noise is ﬁtted to experimental data, the model produces learning effects similar to those found in monkey experiments.

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Functional network reorganization in motor cortex can be explained by reward-modulated Hebbian learning Robert Legenstein1∗ Steven M. [sent-1, score-0.411]

2 It was recently shown that tuning properties of neurons in monkey motor cortex are adapted selectively in order to compensate for an erroneous interpretation of their activity. [sent-4, score-1.145]

3 In particular, it was shown that the tuning curves of those neurons whose preferred directions had been misinterpreted changed more than those of other neurons. [sent-5, score-0.52]

4 This learning rule utilizes neuronal noise for exploration and performs Hebbian weight updates that are modulated by a global reward signal. [sent-7, score-0.39]

5 When the neuronal noise is ﬁtted to experimental data, the model produces learning effects similar to those found in monkey experiments. [sent-10, score-0.446]

6 The monkey’s intended movement velocity vector can be extracted from the ﬁring rates of a group of recorded units by the population vector algorithm, i. [sent-14, score-0.464]

7 1 In [1], this velocity vector was used to control a cursor in a 3D virtual reality environment. [sent-17, score-0.585]

8 The task for the monkey was to move the cursor from the center of an imaginary cube to a target appearing at one of its corners. [sent-18, score-0.809]

9 , the function of recorded neurons is adapted such that control over the new artiﬁcial “limb” is improved [3]. [sent-21, score-0.597]

10 1 1 rates of recorded neurons into cursor movements). [sent-30, score-0.959]

11 When the interpretation was altered for a subset of neurons, the tuning properties of the neurons in this subset changed signiﬁcantly stronger than those of neurons for which the interpretation of the readout system was not changed. [sent-31, score-1.042]

12 Hence, the experiment showed that motor cortical neurons can change their activity speciﬁcally and selectively to compensate for an altered interpretation of their activity within some task. [sent-32, score-1.024]

13 Such adjustment strategy is quite surprising, since it is not clear how the cortical adaption mechanism is able to determine for which subset of neurons the interpretation was altered. [sent-33, score-0.517]

14 Furthermore, there exists evidence for the songbird system that motor variability reﬂects meaningful motor exploration that can support continuous learning [20]. [sent-46, score-0.507]

15 Thus, we show in this study that reward-modulated learning is a possible explaination for experimental results about neuronal tuning changes in monkey pre-motor cortex. [sent-50, score-0.423]

16 2 Learning effects in monkey motor cortex In this section, we brieﬂy describe the experimental results of [1] as well as the network that we used to model learning in motor cortex. [sent-52, score-0.853]

17 Neurons in motor and premotor cortex of primates are broadly tuned to intended arm movement direction [21, 3]. [sent-53, score-0.699]

18 2 This sets the basis for the ability to extract intended arm movement from recorded neuronal activity in in these areas. [sent-54, score-0.597]

19 The tuning curve of a direction tuned neuron is given by its ﬁring rate as a function of movement direction. [sent-55, score-0.442]

20 The preferred direction (PD) pi ∈ R3 of a neuron i is deﬁned as the direction in which the cosine ﬁt to its ﬁring rate is maximal, and the modulation depth is deﬁned as the difference in ﬁring rate between the maximum of the cosine ﬁt and the baseline (mean). [sent-57, score-0.486]

21 The tuning functions of an average of 40 recorded neurons were obtained in the Calibration session where the monkey moved its hand in a center out reaching task. [sent-59, score-0.862]

22 In Control, Perturbation, and Washout sessions the monkey had to perform a cursor control task in a 3D virtual reality environment (see Figure 1B). [sent-62, score-0.774]

23 The cursor was initially positioned in the center of an imaginary cube, a target position on one of the corners of the cube was randomly selected and made visible. [sent-63, score-0.609]

24 When the monkey managed to hit the target position with the cursor or a 3s time period expired, the cursor position was reset to the origin and a new target position was randomly selected from the eight corners of the imaginary cube. [sent-64, score-1.275]

25 In the Control session, the measured PDs were used as dPDs for cursor control. [sent-65, score-0.437]

26 In the Perturbation session, the dPDs of a randomly selected subset of neurons (25% or 50% of the recorded neurons) were altered. [sent-66, score-0.522]

27 This was 2 Arm movement refers to movement of the endpoint of the arm. [sent-67, score-0.39]

28 A set of m neurons project to ntotal noisy neurons in motor cortex. [sent-70, score-1.109]

29 The monkey arm movement was modeled by a ﬁxed linear mapping from the activities of the modeled motor cortex neurons to the 3D velocity vector of the monkey arm. [sent-71, score-1.742]

30 A subset of n neurons in the simulated motor cortex was recorded for cursor control. [sent-72, score-1.326]

31 The cursor velocity was given by the population vector. [sent-73, score-0.573]

32 B) The task was to move the cursor from the center of an imaginary cube to one of its eight corners. [sent-74, score-0.529]

33 The measured PDs were used for cursor control in the subsequent Washout session. [sent-78, score-0.483]

34 In the Perturbation session, neurons adapted their ﬁring behavior to compensate for the altered dPDs. [sent-79, score-0.563]

35 The authors observed differential effects of learning for the two groups of non-rotated neurons and rotated neurons. [sent-80, score-0.741]

36 Rotated neurons tended to shift their PDs in the direction of dPD rotation, thus compensating for the perturbation. [sent-81, score-0.558]

37 We refer to this differential behavior of rotated and non-rotated neurons as the “credit assignment effect”. [sent-83, score-0.744]

38 The input population modeled those neurons which provide input to the neurons in motor cortex. [sent-86, score-1.199]

39 It consisted of m = 100 neurons with activities x1 (t), . [sent-87, score-0.541]

40 Another population modeled neurons in motor cortex which receive inputs from the input population. [sent-91, score-0.858]

41 It consisted of ntotal = 340 neurons with activities s1 (t), . [sent-92, score-0.591]

42 3 All modeled motor cortex neurons were used to determine the monkey arm movement in our model. [sent-96, score-1.312]

43 A small number of them (n = 40) modeled recorded neurons used for cursor control. [sent-97, score-0.993]

44 The total synaptic input ai (t) for neuron i at time t was modeled as a noisy weighted sum of its inputs: m ai (t) = wij xj (t) + ξi (t), ξi (t) drawn from distribution D(ν), (1) j=1 where wij is the synaptic efﬁcacy from input neuron j to neuron i. [sent-102, score-0.973]

45 Input neurons may be situated in extracortical areas, in other cortical areas, or even in motor cortex itself. [sent-110, score-0.781]

46 Task model: We modeled the cursor control task as shown in Figure 1B. [sent-116, score-0.517]

47 Eight possible cursor target positions were located at the corners of a unit cube in 3D space which had its center at the origin of the coordinate system. [sent-117, score-0.568]

48 At each time step t the desired direction of cursor movement y∗ (t) was computed from the current cursor and target position. [sent-118, score-1.244]

49 From the desired movement direction y∗ (t), the activities x1 (t), . [sent-120, score-0.408]

50 , xm (t) of the neurons that provide input to the motor cortex neurons were computed and the activities s1 (t), . [sent-123, score-1.273]

51 , sn (t) of the recorded neurons were used to determine the cursor velocity via their population activity vector (see below). [sent-126, score-1.185]

52 In order to model the cursor control experiment, we had to determine the PDs of recorded neurons. [sent-127, score-0.581]

53 In monkeys, the transformation from motor cortical activity to arm movements involves a complicated system of several synaptic stages. [sent-129, score-0.624]

54 Experimental ﬁndings suggest that monkey arm movements can be predicted quite well by a linear model based on the activities of a small number of motor cortex neurons [3]. [sent-131, score-1.237]

55 We therefore assumed that the direction of the monkey arm movement yarm (t) at time t can be modeled in a linear way, using the activities of the total population of the ntotal cortical neurons s1 (t), . [sent-132, score-1.304]

56 With the transformation from motor cortex neurons to monkey arm movements being deﬁned, the input to the network for a given desired direction y∗ should be chosen such that motor cortex neurons produce a monkey arm movement close to the desired movement direction. [sent-136, score-2.874]

57 As described above, a subset of the motor cortex population was chosen to model recorded neurons that were used for cursor control. [sent-139, score-1.335]

58 , n}, we determined the preferred direction pi ∈ R3 as well as the baseline activity βi and the modulation depth αi by ﬁtting a cosine tuning on the basis of simulated monkey arm movements [1, 23]. [sent-143, score-0.849]

59 The dPDs were then used to determine the movement velocity y(t) of the cursor by the population vector algorithm [1, 2, 23]. [sent-145, score-0.768]

60 3 Adaptation with an online learning rule Adaptation of synaptic efﬁcacies wij from input neurons to neurons in motor cortex is necessary ˜ if the actual decoding PDs pi do not produce optimal cursor trajectories. [sent-147, score-1.955]

61 Then for some input x(t), the movement of the cursor is not in the desired direction y∗ (t). [sent-152, score-0.786]

62 The weights wij should therefore be adapted such that at every time step t the direction of movement y(t) is close to the desired direction y∗ (t). [sent-153, score-0.506]

63 We can quantify the angular match Rang (t) at time t by the cosine of the angle between movement direction y(t) and T ∗ y(t) y (t) desired direction y∗ (t): Rang (t) = ||y(t)||·||y∗(t)|| . [sent-154, score-0.507]

64 This measure has a value of 1 if the cursor moves exactly in the desired direction, it is 0 if the cursor moves perpendicular to the desired direction, and it is -1 if the cursor movement is in the opposite direction. [sent-155, score-1.606]

65 In that 4 study, the idea was implemented by learning rules where the weight changes are proportional to the covariance between the reward signal R and some measure of neuronal activity N at the synapse. [sent-158, score-0.396]

66 We investigate in this article a learning rule of this type: ¯ EH rule: ∆wij (t) = η xj (t) [ai (t) − ai (t)] R(t) − R(t) , ¯ (2) ¯ where ai (t) and R(t) denote the low-pass ﬁltered version of ai (t) and R(t) with an exponential ¯ kernel4 . [sent-162, score-0.379]

67 The deviation of the activation from the recent mean ai (t) − ai (t) is an estimate of the exploratory term ξi (t) at time t if the mean ai (t) is ¯ ¯ based on neuron activations j wij xj (t′ ) which are similar to the activation j wij xj (t) at time t. [sent-170, score-0.699]

68 For example, in [6] and [7] binary neurons were used. [sent-179, score-0.424]

69 Instead a temporally ﬁltered version of the neurons activation is used to estimate the noise signal. [sent-183, score-0.519]

70 4 Comparison with experimentally observed learning effects In this section, we explore the EH rule (2) in a cursor control task that was modeled to closely match the experimental setup in [1]. [sent-185, score-0.684]

71 In monkey experiments, perturbation of decoding PDs lead to retuning of PDs with the above described credit assignment effect [1]. [sent-187, score-0.512]

72 We note that in all simulations with the EH rule, the input activities xj (t) were scaled in such a way that the output of the neuron at time t could be interpreted directly as the ﬁring rate of the neuron 4 5 ¯ ¯ We used ai (t) = 0. [sent-194, score-0.481]

73 B) PD shifts drawn on the unit sphere (arbitrary units) for non-rotated (black traces) and rotated (light cyan traces) neurons from their initial values (light) to their values after training (dark, these PDs are connected by the shortest path on the unit sphere). [sent-205, score-0.695]

74 The PDs of rotated neurons are consistently rotated in order to compensate for the perturbation. [sent-208, score-1.003]

75 With such scaling, we obtained output values of the neurons without the exploratory signal in the range of 0 to 120Hz with a roughly exponential distribution. [sent-214, score-0.52]

76 To constrain this parameter, η was chosen such that the performance in the 25% perturbation task approximately matched the monkey performance. [sent-216, score-0.395]

77 In the ﬁrst set of simulations, a random set of 25% of recorded neurons were rotated neurons in Perturbation sessions. [sent-218, score-1.217]

78 In the second set of simulations, we chose 50 % of the recorded neurons to be rotated. [sent-219, score-0.522]

79 The shifts in PDs of recorded neurons induced by training in 20 independent trials were compiled and analyzed separately for rotated neurons and non-rotated neurons. [sent-222, score-1.217]

80 In the simulated 25% perturbation 6 experiment, the mean shift of the PD for rotated neurons was 8. [sent-224, score-0.958]

81 In the simulated 50% perturbation experiment, the mean shift of the PD for rotated neurons was 18. [sent-239, score-0.958]

82 In the simulated 50% perturbation experiment, the mean shift of the PD of rotated neurons (non-rotated neurons) was 12. [sent-257, score-0.958]

83 In the monkey experiment, training in the Perturbation session also induced in a decrease of the modulation depth of rotated neurons. [sent-265, score-0.656]

84 This resulted in a decreased contribution of these neurons to the cursor movement. [sent-266, score-0.861]

85 Modulation depths for non-rotated neurons increased on average by 2. [sent-271, score-0.477]

86 7 Thus, the relative contribution of rotated neurons on cursor movement decreased. [sent-277, score-1.327]

87 For small exploration levels, PDs changed only slightly and the difference in PD change between rotated and non-rotated neurons was small, while for large noise levels, PD change differences can be quite drastic. [sent-280, score-0.863]

88 The neurons in the network after training are subject to the same amount of noise as the neurons in the network before training, but the angular match after training shows much less ﬂuctuation than before training. [sent-283, score-1.06]

89 We quantiﬁed this observation by computing the standard deviation of the angle between the cursor velocity vector and the desired movement direction for 100 randomly drawn noise samples. [sent-285, score-0.93]

90 Angular match Rang (t) of the cursor movements in one reaching trial before (gray) and after (black) learning as a function of the time since the target was ﬁrst made visible. [sent-296, score-0.581]

91 Using the re-aiming strategy, the monkey compensates for perturbations by aiming for a virtual target located in the direction that offsets the visuomotor rotation. [sent-301, score-0.384]

92 A reduction of modulation depths of rotated neurons was also identiﬁed in the experimentals. [sent-305, score-0.839]

93 Rotated neurons shifted their PDs more than the non-rotated population in the experiments. [sent-307, score-0.482]

94 This modeling study therefore suggests that all three elements can be explained by a single synaptic adaptation strategy that relies on noisy neuronal activity and visual feedback that is made accessible to all synapses in the network by a global reward signal. [sent-310, score-0.495]

95 The output of non-rotated neurons is consistent with the interpretation of the readout system. [sent-313, score-0.494]

96 On the other hand, if the output of a rotated neuron is radically different, this will often improve performance. [sent-315, score-0.392]

97 Thus, this study shows that reward-modulated learning can explain detailed experimental results about neuronal adaptation in motor cortex and therefore suggests that reward-modulated learning is an essential plasticity mechanism in cortex. [sent-318, score-0.562]

98 The EH rule (2) is one particularly simple instance of such rules that exploits temporal continuity of inputs and an exploration signal - a signal which would show up as “noise” in neuronal recordings. [sent-322, score-0.389]

99 Primate motor cortex and free arm movements to visual targets in three- dimensional space. [sent-361, score-0.495]

100 coding of the direction of movement by a neuronal population. [sent-363, score-0.383]

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same-paper 1 1.0000004 99 nips-2009-Functional network reorganization in motor cortex can be explained by reward-modulated Hebbian learning

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Abstract: One of the central problems in neuroscience is reconstructing synaptic connectivity in neural circuits. Synapses onto a neuron can be probed by sequentially stimulating potentially pre-synaptic neurons while monitoring the membrane voltage of the post-synaptic neuron. Reconstructing a large neural circuit using such a “brute force” approach is rather time-consuming and inefficient because the connectivity in neural circuits is sparse. Instead, we propose to measure a post-synaptic neuron’s voltage while stimulating sequentially random subsets of multiple potentially pre-synaptic neurons. To reconstruct these synaptic connections from the recorded voltage we apply a decoding algorithm recently developed for compressive sensing. Compared to the brute force approach, our method promises significant time savings that grow with the size of the circuit. 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This method can be scaled to reconstruct multiple synaptic connections onto one neuron by combining intracellular recordings from the postsynaptic neuron with photo-activation of pre-synaptic neurons using glutamate uncaging [913] or channelrhodopsin [14, 15], or with multi-electrode arrays [16, 17]. Neurons are sequentially stimulated to fire action potentials by scanning a laser beam (or electrode voltage) over a brain slice, while synaptic weights are measured by recording post-synaptic voltage. Although sequential excitation of single potentially pre-synaptic neurons could reveal connectivity, such a “brute force” approach is inefficient because the connectivity among neurons is sparse. Even among nearby neurons in the cerebral cortex, the probability of connection is only about ten percent [3-8]. Connection probability decays rapidly with the 1 distance between neurons and falls below one percent on the scale of a cortical column [3, 8]. Thus, most single-neuron stimulation trials would result in zero response making the brute force approach slow, especially for larger circuits. Another drawback of the brute force approach is that single-neuron stimulation cannot be combined efficiently with methods allowing parallel recording of neural activity, such as calcium imaging [18-22], voltage-sensitive dyes [23-25] or multi-electrode arrays [17, 26]. As these techniques do not reliably measure sub-threshold potential but report only spiking activity, they would reveal only the strongest connections that can drive a neuron to fire [2730]. Therefore, such combination would reveal only a small fraction of the circuit. We propose to circumvent the above limitations of the brute force approach by stimulating multiple potentially pre-synaptic neurons simultaneously and reconstructing individual connections by using a recently developed method called compressive sensing (CS) [31-35]. In each trial, we stimulate F neurons randomly chosen out of N potentially pre-synaptic neurons and measure post-synaptic activity. Although each measurement yields only a combined response to stimulated neurons, if synaptic inputs sum linearly in a post-synaptic neuron, one can reconstruct the weights of individual connections by using an optimization algorithm. Moreover, if the synaptic connections are sparse, i.e. only K << N potentially pre-synaptic neurons make synaptic connections onto a post-synaptic neuron, the required number of trials M ~ K log(N/K), which is much less than N [31-35]. The proposed method can be used even if only spiking activity is available. Because multiple neurons are driven to fire simultaneously, if several of them synapse on the post-synaptic neuron, they can induce one or more spikes in that neuron. As quantized spike counts carry less information than analog sub-threshold voltage recordings, reconstruction requires a larger number of trials. 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We represent synaptic weights as components of a column vector x, where zeros represent non-existing connections. Each row in the stimulation matrix A represents a trial, ones indicating neurons driven to spike once and zeros indicating non-spiking neurons. The number of rows in the stimulation matrix A is equal to the number of trials M. The column vector y represents M measurements of membrane voltage obtained by an intra-cellular recording from the post-synaptic neuron: y = Ax. (1) In order to recover individual synaptic weights, Eq. (1) must be solved for x. RC (or STA) solution to this problem is x = (ATA)-1AT y, which minimizes (y-Ax)2 if M>N. In the case M << N for a sparse circuit. In this section we search computationally for the minimum number of trials required for exact reconstruction as a function of the number of non-zero synaptic weights K out of N potentially pre-synaptic neurons. First, note that the number of trials depends on the number of stimulated neurons F. If F = 1 we revert to the brute force approach and the number of measurements is N, while for F = N, the measurements are redundant and no finite number suffices. As the minimum number of measurements is expected to scale as K logN, there must be an optimal F which makes each measurement most informative about x. To determine the optimal number of stimulated neurons F for given K and N, we search for the minimum number of trials M, which allows a perfect reconstruction of the synaptic connectivity x. For each F, we generate 50 synaptic weight vectors and attempt reconstruction from sequentially increasing numbers of trials. The value of M, at which all 50 recoveries are successful (up to computer round-off error), estimates the number of trial needed for reconstruction with probability higher than 98%. By repeating this procedure 50 times for each F, we estimate the mean and standard deviation of M. We find that, for given N and K, the minimum number of trials, M, as a function of the number of stimulated neurons, F, has a shallow minimum. As K decreases, the minimum shifts towards larger F because more neurons should be stimulated simultaneously for sparser x. For the explored range of simulation parameters, the minimum is located close to 0.1N. Next, we set F = 0.1N and explore how the minimum number of measurements required for exact reconstruction depends on K and N. Results of the simulations following the recipe described above are shown in Figure 3a. As expected, when x is sparse, M grows approximately linearly with K (Figure 3b), and logarithmically with N (Figure 3c). N = 1000 180 25 160 20 140 120 15 100 10 80 5 150 250 400 650 1000 Number of potential connections (N) K = 30 220 200 (a) 200 220 Number of measurements (M) 30 Number of measurements (M) Number of actual connections (K) Number of necessary measurements (M) (b) 180 160 140 120 100 80 5 10 15 20 25 30 Number of actual connections (K) 210 (c) 200 190 180 170 160 150 140 130 120 2 10 10 3 Number of potential connections (N) Figure 3: a) Minimum number of measurements M required for reconstruction as a function of the number of actual synapses, K, and the number of potential synapses, N. b) For given N, we find M ~ K. c) For given K, we find M ~ logN (note semi-logarithmic scale in c). 4 R o b ust nes s o f re con st r uc t io n s t o noi se a n d v io la tio n o f si m pli fy in g a ss umpt io n s To make our simulation more realistic we now take into account three possible sources of noise: 1) In reality, post-synaptic voltage on a given synapse varies from trial to trial [4, 5, 46-52], an effect we call synaptic noise. Such noise detrimentally affects reconstructions because each row of A is multiplied by a different instantiation of vector x. 2) Stimulation of neurons may be imprecise exciting a slightly different subset of neurons than intended and/or firing intended neurons multiple times. We call this effect stimulation noise. Such noise detrimentally affects reconstructions because, in its presence, the actual measurement matrix A is different from the one used for recovery. 3) A synapse may fail to release neurotransmitter with some probability. Naturally, in the presence of noise, reconstructions cannot be exact. We quantify the 4 reconstruction x − xr l2 = error ∑ N i =1 by the normalized x − xr l2–error l2 / xl , where 2 ( xi − xri ) 2 . We plot normalized reconstruction error in brute force approach (M = N = 500 trials) as a function of noise, as well as CS and RC reconstruction errors (M = 200, 600 trials), Figure 4. 2 0.9 Normalized reconstruction error ||x-x|| /||x|| 1 r 2 For each noise source, the reconstruction error of the brute force approach can be achieved with 60% fewer trials by CS method for the above parameters (Figure 4). For the same number of trials, RC method performs worse. Naturally, the reconstruction error decreases with the number of trials. The reconstruction error is most sensitive to stimulation noise and least sensitive to synaptic noise. 1 (a) 1 (b) 0.9 0.8 (c) 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0.7 RC: M=200 RC: M=600 Brute force method: M=500 CS: M=200 CS: M=600 0.6 0.5 0 0.05 0.1 0.15 Synaptic noise level 0.2 0.25 0 0.05 0.1 0.15 Stimulation noise level 0.2 0.25 0 0 0.05 0.1 0.15 0.2 0.25 Synaptic failure probability Figure 4: Impact of noise on the reconstruction quality for N = 500, K = 30, F = 50. a) Recovery error due to trial-to-trial variation in synaptic weight. The response y is calculated using the synaptic connectivity x perturbed by an additive Gaussian noise. The noise level is given by the coefficient of variation of synaptic weight. b) Recovery error due to stimulation noise. The matrix A used for recovery is obtained from the binary matrix used to calculate the measurement vector y by shifting, in each row, a fraction of ones specified by the noise level to random positions. c) Recovery error due to synaptic failures. The detrimental effect of the stimulation noise on the reconstruction can be eliminated by monitoring spiking activity of potentially pre-synaptic neurons. By using calcium imaging [18-22], voltage-sensitive dyes [23] or multi-electrode arrays [17, 26] one could record the actual stimulation matrix. Because most random matrices satisfy the reconstruction requirements [31, 34, 35], the actual stimulation matrix can be used for a successful recovery instead of the intended one. If neuronal activity can be monitored reliably, experiments can be done in a different mode altogether. Instead of stimulating designated neurons with high fidelity by using highly localized and intense light, one could stimulate all neurons with low probability. Random firing events can be detected and used in the recovery process. The light intensity can be tuned to stimulate the optimal number of neurons per trial. Next, we explore the sensitivity of the proposed reconstruction method to the violation of simplifying assumptions. First, whereas our simulation assumes that the actual number of connections, K, is known, in reality, connectivity sparseness is known a priori only approximately. Will this affect reconstruction results? In principle, CS does not require prior knowledge of K for reconstruction [31, 34, 35]. For the CoSaMP algorithm, however, it is important to provide value K larger than the actual value (Figure 5a). Then, the algorithm will find all the actual synaptic weights plus some extra non-zero weights, negligibly small when compared to actual ones. Thus, one can provide the algorithm with the value of K safely larger than the actual one and then threshold the reconstruction result according to the synaptic noise level. Second, whereas we assumed a linear summation of inputs [53], synaptic integration may be 2 non-linear [54]. We model non-linearity by setting y = yl + α yl , where yl represents linearly summed synaptic inputs. Results of simulations (Figure 5b) show that although nonlinearity can significantly degrade CS reconstruction quality, it still performs better than RC. 5 (b) 0.45 0.4 Actual K = 30 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 10 20 30 40 50 60 Normalized reconstruction error ||x-x||2/||x|| r 2 Normalized reconstrcution error ||x-x||2/||x|| r 2 (a) 0.9 0.8 0.7 CS RC 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.15-0.12-0.09-0.06-0.03 0 0.03 0.06 0.09 0.12 0.15 Relative strength of the non-linear term α × mean(y ) K fed to CoSaMP l Figure 5: Sensitivity of reconstruction error to the violation of simplifying assumptions for N = 500, K = 30, M = 200, F = 50. a) The quality of the reconstruction is not affected if the CoSaMP algorithm is fed with the value of K larger than actual. b) Reconstruction error computed in 100 realizations for each value of the quadratic term relative to the linear term. 5 Ma pp i ng sy na pti c inp ut s o nto a n e uro na l po pu la tio n Until now, we considered reconstruction of synaptic inputs onto one neuron using subthreshold measurements of its membrane potential. In this section, we apply CS to reconstructing synaptic connections onto a population of potentially post-synaptic neurons. Because in CS the choice of stimulated neurons is non-adaptive, by recording from all potentially post-synaptic neurons in response to one sequence of trials one can reconstruct a complete feedforward network (Figure 6). (a) x p(y=1) A Normalized reconstruction error ||x/||x||2−xr/||xr||2||2 y (b) (c) 100 (d) 500 700 900 1100 Number of spikes 1 STA CS 0.8 0.6 0.4 0.2 0 1000 0 300 3000 5000 7000 9000 Number of trials (M) Ax Figure 6: Mapping of a complete feedforward network. a) Each post-synaptic neuron (red) receives synapses from a sparse subset of potentially pre-synaptic neurons (blue). b) Linear algebra representation of the experimental protocol. c) Probability of firing as a function of synaptic current. d) Comparison of CS and STA reconstruction error using spike trains for N = 500, K = 30 and F = 50. Although attractive, such parallelization raises several issues. First, patching a large number of neurons is unrealistic and, therefore, monitoring membrane potential requires using different methods, such as calcium imaging [18-22], voltage sensitive dyes [23-25] or multielectrode arrays [17, 26]. As these methods can report reliably only spiking activity, the measurement is not analog but discrete. Depending on the strength of summed synaptic inputs compared to the firing threshold, the postsynaptic neuron may be silent, fire once or multiple times. As a result, the measured response y is quantized by the integer number of spikes. Such quantized measurements are less informative than analog measurements of the sub-threshold membrane potential. In the extreme case of only two quantization levels, spike or no spike, each measurement contains only 1 bit of information. Therefore, to achieve reasonable reconstruction quality using quantized measurements, a larger number of trials M>>N is required. We simulate circuit reconstruction from spike recordings in silico as follows. First, we draw synaptic weights from an experimentally motivated distribution. Second, we generate a 6 random stimulation matrix and calculate the product Ax. Third, we linear half-wave rectify this product and use the result as the instantaneous firing rate for the Poisson spike generator (Figure 6c). We used a rectifying threshold that results in 10% of spiking trials as typically observed in experiments. Fourth, we reconstruct synaptic weights using STA and CS and compare the results with the generated weights. We calculated mean error over 100 realizations of the simulation protocol (Figure 6d). Due to the non-linear spike generating procedure, x can be recovered only up to a scaling factor. We propose to calibrate x with a few brute-force measurements of synaptic weights. Thus, in calculating the reconstruction error using l2 norm, we normalize both the generated and recovered synaptic weights. Such definition is equivalent to the angular error, which is often used to evaluate the performance of STA in mapping receptive field [37, 55]. Why is CS superior to STA for a given number of trials (Figure 6d)? Note that spikeless trials, which typically constitute a majority, also carry information about connectivity. While STA discards these trials, CS takes them into account. In particular, CoSaMP starts with the STA solution as zeroth iteration and improves on it by using the results of all trials and the sparseness prior. 6 D i s c uss ion We have demonstrated that sparse feedforward networks can be reconstructed by stimulating multiple potentially pre-synaptic neurons simultaneously and monitoring either subthreshold or spiking response of potentially post-synaptic neurons. When sub-threshold voltage is recorded, significantly fewer measurements are required than in the brute force approach. Although our method is sensitive to noise (with stimulation noise worse than synapse noise), it is no less robust than the brute force approach or RC. The proposed reconstruction method can also recover inputs onto a neuron from spike counts, albeit with more trials than from sub-threshold potential measurements. This is particularly useful when intra-cellular recordings are not feasible and only spiking can be detected reliably, for example, when mapping synaptic inputs onto multiple neurons in parallel. For a given number of trials, our method yields smaller error than STA. The proposed reconstruction method assumes linear summation of synaptic inputs (both excitatory and inhibitory) and is sensitive to non-linearity of synaptic integration. Therefore, it is most useful for studying connections onto neurons, in which synaptic integration is close to linear. On the other hand, multi-neuron stimulation is closer than single-neuron stimulation to the intrinsic activity in the live brain and can be used to study synaptic integration under realistic conditions. In contrast to circuit reconstruction using intrinsic neuronal activity [56, 57], our method relies on extrinsic stimulation of neurons. Can our method use intrinsic neuronal activity instead? We see two major drawbacks of such approach. First, activity of non-monitored presynaptic neurons may significantly distort reconstruction results. Thus, successful reconstruction would require monitoring all active pre-synaptic neurons, which is rather challenging. Second, reliable reconstruction is possible only when the activity of presynaptic neurons is uncorrelated. Yet, their activity may be correlated, for example, due to common input. 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