Animals must respond selectively to specific combinations of salient environmental stimuli in order to survive in complex environments. A task with these features, biconditional discrimination, requires responses to select pairs of stimuli that are opposite to responses to those stimuli in another combination. We investigate the characteristics of synaptic plasticity and network connectivity needed to produce stimulus-pair neural responses within randomly connected model networks of spiking neurons trained in biconditional discrimination. Using reward-based plasticity for synapses from the random associative network onto a winner-takes-all decision-making network representing perceptual decision-making, we find that reliably correct decision making requires upstream neurons with strong stimulus-pair selectivity. By chance, selective neurons were present in initial networks; appropriate plasticity mechanisms improved task performance by enhancing the initial diversity of responses. We find long-term potentiation of inhibition to be the most beneficial plasticity rule by suppressing weak responses to produce reliably correct decisions across an extensive range of networks.
Most environmental stimuli, to which an animal must develop an appropriate response, comprise multiple features and sub-features that are common to many other stimuli. Since these other stimuli could engender an alternative response by the animal, it is essential that an animal is able to recognize specific combinations of stimulus features in order to distinguish and respond effectively to differing stimuli that share many features. The simplest step in the formation of specific responses to complex stimuli is the ability to combine two inputs and produce a response distinct from either input alone or other input pairings. Associative learning is necessary for an animal to recognize that at least two previously unrelated objects or events comprise a composite stimulus that requires a specific response [1], [2], [3], [4], [5]. Some of the most difficult associative learning processes involve tasks that utilize exclusive-or, XOR, logic (Figure 1A). Associative learning tasks that employ XOR logic include pair-associative learning [3], [6], transitive inference [7], and biconditional discrimination tasks [8], [9], among others. These tasks vary in design and sensory modality, but they all share one requirement, the development of stimulus-pair selectivity to solve the task. Rats and monkeys require extensive training [2], [10] to perform well in such tasks. The difficulty in XOR tasks (Figure 1A) arises from the requirement for an animal to produce a response to stimulus-pairs (e. g. A+B) selectively, in a manner that differs from its response to the individual stimuli that comprise them (e. g. A or B). For example, in biconditional discrimination (Figure 1A) [9], if the animal learns to respond to one member of the stimulus-pair (e. g. B from A+B) then while it will respond correctly to stimulus-pair A+B it would respond incorrectly to C+B. Thus, in biconditional discrimination, as in other tasks based on XOR logic, successful decision-making requires responses selective to stimulus-pairs (e. g. A+B vs. C+B). The results of our studies based on the biconditional discrimination task can be applied to a number of associative learning tasks that employ XOR logic such as visual association [3], [6], transitive inference tasks [11], and many others [4]. What remains unclear in these tasks is how the requisite stimulus-pair representations form. Here we investigate how cells responsive to specific conjunctions of stimuli form, by examining what synaptic plasticity rules can generate stimulus-pair specificity within a randomly connected network of spiking neurons and compare with the likelihood of their initial chance occurrence [12]. Our work shares some similarities to a previous computational study [13], which produced associations between individual, temporally separated stimuli in structured networks. However, our focus is on the general role of network connectivity [12] and synaptic plasticity rules described in vitro, necessary to solve multiple tasks requiring pair-associative learning. We study the well-known correlation-based mechanism for changing excitatory synaptic strengths, spike-timing-dependent plasticity (STDP) [14], [15] as well as a more recent formulation, triplet STDP [16]. Triplet STDP is distinguished from standard STDP through its rate dependence – favoring potentiation over depression as overall rate increases. Standard STDP determines the sign of plasticity from the relative times of each single presynaptic and single postsynaptic spike pair but fails to replicate such rate dependence. The higher order spike interactions included in triplet STDP fit recent in vitro data better [16], [17], [18], [19], as well as the observed rate dependence of more classic experiments data [17], [18], [20], [21], [22] while maintaining standard STDP observations [16]. Thus, we incorporate a recent computational model of triplet STDP to determine how its affects the network differently from that of standard STDP. Recent modeling studies of recurrent networks undergoing STDP suggest the plasticity mechanism could be detrimental in the formation of pair-specific responses for two reasons. First, the competition among inputs to a single cell inherent in STDP [23] could lead single cells to become responsive to a single stimulus, or the complete network to respond to only one stimulus-pair [24]. Alternatively, plasticity among the excitatory connections can lead to a phenomenon termed attractor accretion in recent work [25] whereby cells associate with multiple stimulus-pairs. Such over-association would be detrimental when a specific stimulus-pair response is necessary, but has been shown to be useful when generalization is necessary [25]. Thus, in addition to excitatory plasticity, we model a recently described form of inhibitory plasticity [26], [27] long-term potentiation of inhibition (LTPi), which produces an increase in strength of inhibitory connections to excitatory cells if the inhibitory cell spikes while the excitatory cell is depolarized but not spiking. We study how these excitatory and inhibitory plasticity rules operate in conjunction with multiplicative postsynaptic scaling, a mechanism for homeostasis [28]. We make minimal assumptions regarding network structure by studying networks with random afferent projections and random recurrent connections. To demonstrate the robustness of learning rules, we study them in a variety of networks, with differing levels of sparseness, excitability and degree of correlation in the connections from input groups that respond to individual stimuli. We define a measure of pair selectivity at the neuronal level, and measure the distribution of selectivity across cells before and after training. When comparing multiple networks, we use the mean of the stimulus-pair selectivity across cells. In order to determine whether or not the information about stimulus-pairs within a given associative network is sufficient to produce a reliable behavioral response, we train a binary winner-takes-all (WTA) network, whose inputs are obtained from our associative network. The WTA network serves as a model for perceptual decision-making [29], [30]. Its afferent synapses are modified by a Dopamine (DA) reward-based plasticity rule that, in principle, can lead it to produce responses that maximize reward [31]. We found that in many cases, both standard STDP and triplet STDP produced lower selectivity to stimulus pairs and less reliable decision-making performance than found in the network before learning. This limitation on the ability of STDP to produce pair-selective cells arose from potentiation of synaptic connections between cells, which were initially selectively responsive to different stimulus pairs, but gained responses to the stimulus pair favored by the connected cell. We term this undesirable phenomenon of losing selectivity through the gaining of extra responses as ‘over-associativity. ’ Over-associativity was prevented by LTPi, which could produce cross-inhibition. Networks trained with LTPi alone or in combination with STDP produced reliable decision-making across the largest range of networks tested in this study. Thus, these results demonstrate a valuable role for this recently discovered form of inhibitory plasticity.
Throughout this paper we describe how learning rules affect stimulus-pair selectivity. Stimulus-pair selectivity can be plainly stated as how responsive a neuron' s firing rate is to one stimulus-pair (e. g. A+B) over all other stimulus-pairs (for a formal definition, see the experimental procedures). Any cell responding equally to all four stimulus-pairs is least selective (giving a measure of 0) while any cell responding to a single stimulus-pair is the most selective (giving a measure of 3). A concrete example of a single neuron (Figure 2A, B) is useful for understanding the selectivity metric. Initially, the neuron is approximately equally responsive (as measured by the number of spikes produced) to each stimulus-pair (giving a measure of near 0) (Figure 2A); however after training with LTPi and triplet STDP, the neuron becomes selective to only stimulus-pair A+B (giving a measure of 3), maintaining its initial firing rate in response to the combination A+B, despite the pruning of other stimulus-pair responses (Figure 2B). Non-linearity is necessary for cells to generate stimulus-pair selectivity greater than one, however selective it is to individual inputs. For example, a cell responding linearly to inputs only from stimulus “A” would fire at a rate, rA, to stimulus-pairs “A+B” and “A+D” and at a rate of zero to stimulus-pairs “C+B” and “C+D”, producing a stimulus-pair selectivity of 1. Such a cell could not help in the task. Similarly, a cell responding linearly to the individual inputs “A” and “B”, firing at rate rA+rB, to the pair “A+B”, at rate rA, to pair “A+D”, at rate rB, to pair “C+B” and a rate of zero to pair “C+D” would also have stimulus-pair selectivity of one. Such cells are also unlikely be unhelpful in training an XOR task, since they produce equal numbers of spikes for the two desired responses producing equal drive to the decision-making network (spikes fired to stimulus-pairs “A+B” and “C+D” equals spikes fired to “C+B” and “C+D”). The excess spikes of a single cell in response to a stimulus-pair are likely to be swamped by noise, unless other cells respond similarly to the same stimulus-pair. Thus, to assess how well the network as a whole produces pair-selectivity, we measure the distribution of selectivity across all excitatory cells before and after learning (Figure 2C). We assess network responses by examining how the final distribution compares to the initial distribution. Figure 2C provides an example of a population that increased its selectivity following training, as seen by the rightward shift in the final overall distribution, along with many cells reaching the maximum selectivity value of 3. Hereafter, we use the mean of the distribution across cells as a measure of the network' s pair selectivity (Figure 2D). In the Supplementary Information (Figure S2) we describe how well measures of pair-selectivity correlate with our measure of behavioral performance described in a later section of the results.
In this work, we have demonstrated how local cell-specific rules [14], [16], [27], [34], [35] affect global network function to produce the stimulus-pair selectivity as a solution to cognitive tasks with the underpinnings of exclusive-or, XOR, logic [36], [37]. The qualitative robustness of our results, demonstrated by modeling a broad range of networks and conditions, extends these findings broadly, showing they are not the result of a specific set of hand-tuned parameters. Our associative network starts as a completely general one, but becomes sculpted via the paired stimuli it receives to maximally respond to those stimulus-pairs. Combining the unsupervised learning of the associative network with the reward-based learning of connections to the decision-making layer, leads to a system that learns to respond to salient stimuli (i. e. those that determine reward) in the environment. The condition of our network before learning is based on the minimal assumption of random connectivity, yet with appropriate plasticity rules, the functional structure can evolve to allow a fundamental cognitive task to be solved. It is likely that the base structure of specific areas of the brain – such as the structured connectivity typical of cortex [38] – provides an advantage in solving relevant tasks. Thus, future investigations can be illuminating of the effect on learning of other structures that more closely resemble cortex for initial connectivity, such as a small-world network [39], [40]. We find that homeostasis is essential within our networks, since all the plasticity rules (including standard STDP) can be unstable in a sparse, recurrent network. With homeostasis, we find that firing rates converge to a steady state value, though it is not the same as the goal rate dictated by homeostasis. That is, when multiple plasticity mechanisms combine simultaneously within a network, the steady state (i. e. the final stable) activity pattern differs from that of any single plasticity mechanism acting alone. One question we investigated is whether sparse or dense activity of cells is beneficial for producing solutions to paired-stimulus association tasks. The argument for sparse firing runs as follows. If one input is insufficient to cause a cell to fire, then cells only fire when two of their inputs are active. If input connections are highly sparse, then given only 4 stimuli are used, the chances of any cell receiving inputs from greater than 2 stimuli become negligible. Thus, any cell receiving multiple inputs and being able to become active does so when its unique stimulus-pair is present. Indeed, we did find that as networks became sparser, the number of active cells became lower, but the selectivity of those active cells became higher. Nevertheless, when measuring decision-making performance, these networks were unreliable, essentially because the downstream neurons received too little input from such sparse firing to overcome random fluctuations from background activity. Mongillo et al. [13] have shown that when different inputs are non-overlapping, Hebbian plasticity of excitatory synapses alone is sufficient to produce paired associations, even when the stimuli are separated in time. Such a sparse extreme of no overlap is optimal for producing discrete pools of cells, which respond persistently to single stimuli. The paired association corresponded to the synaptic connection from one discrete pool to another. In essence, the initial sparseness led to individual stimulus-specific pools that became homogeneous via intra-pool excitatory plasticity. These properties are not ideal when one stimulus can be paired with multiple other stimuli with the required response dependent on the particular pairing (as with XOR logic). Essentially, A-responsive cells cannot be pooled together if stimulus A combined with stimulus B (e. g. A+B) requires a different response from stimulus A combined with stimulus D (e. g. A+D). The need for heterogeneity is more readily satisfied with randomly overlapping inputs. Recent work by Rigotti et al. [12] suggests that dense activity, found with an input connection probability of ½, would be optimal for solving tasks that incorporate XOR logic. Their work with binary neurons operates in a regime where the non-linearity of saturation at maximal activity is as useful as the non-linearity of the firing threshold at zero activity. In our network, neurons were far from saturation, which is perhaps one reason we did not observe greatest selectivity in these dense networks. However, if cortical neurons operate at a level where input saturation (e. g. via NMDA synapses) is as strong as the firing threshold non-linearity, and if noise fluctuations added to the network by neurons of maximal firing rate are no greater than those at minimal firing rate, then the results based on binary synapses [12] are more relevant than those of our sets of networks. Since the main non-linearity in the responses of our neurons is their firing threshold, optimal selectivity among firing neurons arises if all cells are silent except for those most responsive to a particular stimulus-pair. However, such a limit of sparse activity leads to very few selective cells, which fire at very low rates (mean rate during the stimulus is <3 Hz in the sparsest networks, Figure S1) and are insufficient to drive a reliable response in a downstream decision-making network with typical levels of noise. Thus, denser networks with a greater number of selective cells [12] and higher mean firing rates are beneficial. The optimal network would be based on a trade-off between the total numbers of selective cells, the mean firing rate of those selective cells and how selective they are to particular paired stimuli. No two neurons or initial synapses are the same within our networks. Neurons are individualized by heterogeneity in intrinsic properties (cellular time constant, leak conductance, firing threshold and refractory time), and initial synaptic strengths are drawn from a uniform distribution about a mean. Moreover, sparse, random connectivity, both of inputs and of recurrent connections, ensures that each neuron responds differently to stimuli. That randomness in network structure is a beneficial property [12], [41] for the brain highlights the brain' s nature, as an adaptive, biological organ. We incorporated heterogeneity for two reasons. First, we wanted to more closely approximate biophysical networks and observations of the brain [42], [43], [44], [45], [46]. Second, heterogeneity in the network is critical for its development. Heterogeneity in the connections and cellular properties causes neurons to fire differentially to stimuli. Correlations in the connectivity lead to correlated activity, which plasticity rules act upon [47], [48]. Thus, plasticity can enhance initial diversity of responses to increase the stimulus-pair selectivity of cells. Diversity of neural responses by initial heterogeneity provides an animal with a framework to solve any cognitive task [12]. One can ask whether the role of training is simply the learning of an appropriate motor output from a constant internal representation of the stimuli, or whether training enhances neural responses to those stimuli. In principle, any synaptic plasticity mechanism that increases the initial variability of neural responses should be beneficial in solving XOR-like tasks. Perhaps the most surprising result was that networks with STDP alone, in nearly all cases, failed to produce reliable decisions – indeed performing worse than untrained random networks. The sometimes useful role of STDP in attractor concretion [25] reduced the diversity of responses in our associative network, thus diminishing task performance. We did expect that cross-inhibition – an accentuation of differences in neural responses achieved naturally by LTPi (Figure 3) – could be produced by the combination of Hebbian excitation and a global suppression of activity, through homeostasis. However, while LTPi succeeded over a range of parameters and networks, triplet STDP only succeeded in a finely tuned subset of these parameters. This is likely due to an inherent instability when adjusting the recurrent weights within a single set of cells (the excitatory-to-excitatory connections) in a Hebbian manner. In contrast, the changes wrought by LTPi on excitatory cells do not affect the presynaptic activity of inhibitory cells in the networks we consider here, so overall activity levels are more easily stabilized. While networks modified by LTPi alone had the greatest propensity to generate high selectivity and reliable decisions, LTPi could be added to networks in combination with STDP to increase reliability of decision-making. Given these findings, such a combination of plasticity mechanisms could provide an organism with the most robust learning method by generating a network with strong selectivity and firing rates. Further, in networks that produce short-term memory, it is likely that a mechanism such as triplet STDP of excitatory synapses is needed to generate sufficient recurrent excitation [29], [30], [49]. In summary, heterogeneity of neural responses is essential for producing solutions to certain cognitive tasks [12]. Any plasticity mechanisms that either specifically increase the strongest responses or suppress the weakest responses of cells will enhance any heterogeneity initially present in randomly connected networks and facilitate task performance.
We use leaky integrate-and-fire neurons [50] defined by the leak conductance, gL, synaptic conductances AMPA, NMDA, GABAA, and a refractory conductance. Further, we define the neurons by a resting potential (i. e. leak potential), reset and threshold potential. The threshold potential is dynamic in the sense that it is not a hard threshold; rather, it increases to a maximal value and decreases to a base value as the firing rate increases and decreases respectively. This was added so that at high firing rates the neurons could sustain persistence such as neurons in the decision-making network. We model NMDA' s voltage dependence as described below. LIF neurons had a mean leak reversal potential of VL = −70 mV+/−2. 5 mV, a fixed membrane time constant of τm = 10 ms+/−0. 75 ms and leak conductance of gL = 35 µS+/−1 µS in the standard low threshold regime, and values of gL = 40 µS and gL = 50 µS+/−1 µS in the high threshold regimes. Excitatory neurons had a firing threshold of Vth = −50 mV+/−2 mV, a reset voltage of Vref = −60 mV+/−2 mV, and a refractory time constant of τreset = 2 ms+/−. 25 ms. Inhibitory neurons had a firing threshold of Vth = −50 mV+/−2 mV, a reset voltage of Vref = −60 mV+/−2 mV, and a refractory time constant of τreset = 1 ms+/−. 25 ms. Heterogeneity of these parameters was drawn from uniform distribution with the given ranges. Excitatory LIF neurons had a mean leak reversal potential of VL = −70 mV, membrane time constant of τm = 20 ms, and leak conductance of gL = 35 µS. Excitatory neurons had a firing threshold of Vth = −48 mV, a reset voltage of Vreset = −55 mV, and a refractory time constant of τref = 2 ms. Inhibitory LIF neurons had a mean leak reversal potential of VL = −70 mV, membrane time constant of τm = 10 ms, and leak conductance of gL = 30 µS. Excitatory neurons had a firing threshold of Vth = −50 mV, a reset voltage of Vreset = −55 mV, and a refractory time constant of τref = 1 ms. Synaptic currents were modeled by instantaneous steps after a spike followed by an exponential decay described by the equation below [51]. Recurrent excitatory currents were modeled by AMPA (EAMPA = 0 mV, τAMPA = 2 ms) and NMDA receptors (ENMDA = 0 mV, τNMDA = 100 ms). Inhibitory currents were modeled by GABAA receptors (EGABA = −70 mV, τGABA = 10 ms). NMDA receptors were also defined by the voltage term [52]: Neurons do not have a hard reset; rather we use a refractory conductance with a dynamic behavior in order to mimic a delayed rectifier potassium current described by the synaptic ODE, with an increase in refractory conductance per spike, δgref = 0. 002 µS, refractory time constant τref = 2 ms, and refractory reversal potential Vref = −70 mV. Neurons do not have a hard spike threshold either that reaches a higher depolarized value with each spike. This is important for persistent neural activity in our decision-making network. The max Vth = 150 mV. In order to investigate the robustness of each learning rule, we examined their effects on sets of 25 different networks with each set explored across six network regimes. We examined how the sparseness and correlations of input groups affected both the initial selectivity of a network and how the network responds to each of the synaptic plasticity rules. Input sparseness is defined via the probability of any input group projecting to any given cell. As input connection probability increases, sparseness decreases. We used the following five values for input connection probability: 1/2,1/3,1/5,1/10 and 1/20. We produced different degrees of input correlations by altering the number of independently connected input groups of cells per stimulus, using 2,4, 6,10 or 20 independent groups. Each input group produced independent Poisson spike trains with a mean firing rate defined by: = 480 Hz/Number of Input groups (e. g. 10 input groups of 48 Hz). Correlations weakened progressively as the number of inputs increased due to the increasing number of independent input Poisson spike trains producing the same overall spike rate. Five levels of input sparseness, combined with five different degrees of input correlations led to 25 variant networks in each regime. The goal of the present study is to determine how various forms of synaptic plasticity can operate on an initially randomly connected network (Figure 1B) to produce the functional responses necessary to solve a cognitive task. Thus, our initial network possessed no structure in its afferent connections and in its internal recurrent connections. In the present work we did not alter the random connectivity structure during training, but assessed whether it provided a sufficient substrate for the correlation-based synaptic learning rules to generate functional structure by strengthening and weakening existing synapses. Random connectivity produced cell-to-cell variability since no two cells receive identical inputs. Such heterogeneity of the inputs across cells leads to a network of neurons with diverse stimulus responses. The initial diversity of stimulus responses was typically insufficient to produce the tuned activity needed to solve the behavioral task (Figure 1A), but was essential to provide a basis upon which correlation-based plasticity rules could act differentially. While random connectivity can be thought of as a minimal assumption, in contrast to the fine-tuning needed by many spiking neuron-based models of cognitive tasks, such randomness also provided sufficient variability in responses that in principle the network could be trained to produce specific responses to any pairs of inputs. Excitatory-to-excitatory connections are sparse-random with a probability of 10%. Inhibition is feedforward only, so there are no excitatory-to-inhibitory connections. Inhibitory-to-Inhibitory connections are all-to-all. Finally, Inhibitory-to-excitatory synapses connect randomly with a probability of 25%. Initial synaptic strength is a mean value of W0 = 0. 05+/−50% uniformly about the mean and scales in strength with size. These simulations were carried out with an 400 neuron network with an excitatory∶inhibitory ratio of 4∶1. We examined one set of networks (Figure S8) with recurrent inhibition where the excitatory cells connect with a probability of 25% to any inhibitory cell with a fixed mean strength W0. The decision-making network based on [29] is composed of two excitatory and inhibitory pools of a total 500 neurons with a an excitatory∶inhibitory ratio of 4∶1 and synaptic strength W0 = 0. 25. Connections within each pool are all-to-all. Cross-inhibition is direct from each inhibitory pool to the opposing excitatory pool, which generates winner-take-all activity so that only one pool is stable in the up state (active). Network bistability is generated by strong inhibition and self-excitation. Connections to the decision layer are initially all-to-all from excitatory neurons with a uniform strength of in all trained networks, DW0 = 0. 075. In untrained initial networks, the disparity in firing rates between dense and sparse networks was too large (Figure S1) for a single synaptic strength to effectively drive all networks; thus we used a separate DW0 = 0. 125 for the sparse networks (1/10,1/20). The decision-making network receives a linear ramping input initiating at the start of the cue and continues until the end of the cue where it reaches its maximal value of gurgency = 5 µS at the end of the cue. This input is adapted after the “urgency-gating” model [53], and it ensures that a decision is made each trial. We model two different types of noise. First, we model voltage noise by a Gaussian distribution of zero mean with unit variance and amplitude in the associative layer. Second, we model synaptic conductance noise for the AMPA and GABAA conductance that is drawn from a uniform distribution from zero to 1 with amplitude in the associative layer and amplitude in the decision-making layer. For all connections, changes in synaptic strength are limited to a maximum of 50% per trial, while across all trials; synaptic strength is bounded between zero and 20W0, where W0 is the initial mean synaptic strength. LTPi is modeled after [27]: LTPi occurs when an inhibitory cell' s fires, but the excitatory cell is depolarized and silent. If the excitatory cell is co-active, then there is no change in the synapse strength. We refer to this as a veto effect in our model of LTPi. Any excitatory spike within a window of +/−20 ms for an inhibitory spike will result in a veto. For each inhibitory spike (non-vetoed) the synapse is potentiatiated by idW = 0. 005. LTPi was reported experimentally as a mechanism for increasing (but not decreasing) the strength of inhibitory synapses in cortex [27]. To compensate for the inability of LTPi to depress synapses, we use multiplicative postsynaptic scaling [28] for homeostasis at the inhibitory-to-excitatory synapses. We explicitly model the postsynaptic depolarization required by LTPi by defining a voltage threshold that the postsynaptic excitatory cell must be above in order for potentiation to occur. Because simulation cells do not match experimentally used cells exactly, we explored a wide range of values in Figures S9, S10. In the main body of the paper we used a value of −65 mV, which is 5 mV above the leak reversal. Finally, we include a hard upper bound of inhibitory synaptic strength, such that those cells most strongly inhibited (so being less depolarized as well as not spiking) in practice receive no further potentiation of their inhibitory synapses. We implement STDP using standard methods [23], assuming an exponential window for potentiation following a presynaptic spike at time tpre and for depression following a postsynaptic spike at time tpost, so that the change in connection strength, ΔW, follows: Standard STDP produces changes in synaptic weight whose sign depends only on the relative order of spikes, thus only on the relative order and direction of changes in rate, not on the absolute value of the rate. The LTD amplitude A− was 0. 80, and the LTP amplitude A+ was 1. 20. The LTD time constant, τ−, was 25 ms; the LTP time constant, τ+, was 16 ms. For every spike that updates the synapse the synaptic strength changes by dW = 0. 005. Triplet STDP was modeled after the rule published by Pfister & Gerstner 2006 [16]. Their model includes triplet terms, so that recent postsynaptic spikes boost the amount of potentiation during a “pre-before-post” pairing, while recent presynaptic spikes boost the amount of depression during a “post-before-pre” pairing. Specifically when We use the parameters cited from the full model “all-to-all” cortical parameter sets in the paper. The amplitude terms are doublet LTP, doublet LTD, triplet LTP, and triplet LTD. The time constants we used are τ2+ = 16. 68 ms, τ2− = 33. 7 ms, τy = 125 ms, and τx = 101 ms. These parameters generated an LTD-to-LTP threshold for the postsynaptic cell of 20 Hz, above which uncorrelated Poisson spike trains produce potentiation and below which they produce depression. For every spike that updates the synapse the synaptic strength changes by dW = 0. 005. Synapse stability is maintained by multiplicative postsynaptic scaling [28] that is approximate to the following update on a trial-by-trial basis: The change in synaptic strength, ΔW, is proportional to the difference between the mean rate, , and a goal rate, rgoal, with a rate constant ε. We use the parameters, ε = 0. 01 (inhibitory-to-excitatory synapses), ε = 0. 0001 for input to and recurrent excitatory synapses, excitatory goal rate rgE = 4 Hz in the low goal rate regime and rgE = 8 Hz in the standard regime, inhibitory goal rate rgI = 8 Hz, and inhibitory-to-excitatory goal rate of rgIE = 8 Hz. The goal rates, rgE and rgIE, were heterogeneous about their means with an added 5 Hz random spread from a uniform distribution. Stimulus-pair selectivity defines each neuron' s selectivity for one stimulus-pair over the other three stimulus-pairs. We define this for each excitatory neuron, i, by its maximum firing rate minus the mean response across all stimuli. The stimulus-pair selectivity value is normalized by the neuron' s mean rate so that the rate doesn' t determine selectivity, allowing even low activity neurons that are selective to affect the value. The network value is the mean taken across all excitatory cells unless otherwise stated. This description is defined by the following equation. Simulations were run for 800 trials, and numerically integrated using the Euler-Maruyama method with a time step, dt =. 02 ms. All simulations were run across at least four random instantiations of network structure, cell and synapse heterogeneity, and background noise. Key networks were run for ten random instantiations to ensure robustness. Simulations were written in C++ on Intel Xeon machines. Matlab r2010a was used for data analysis and visualization.
Learning to associate relevant stimuli in our environment is important for survival. For example, identification of an object, such as an edible fruit, may require us to recognize a unique combination of features – color, shape, size – each of which is present in other, perhaps inedible, objects. Thus, how the brain associates distinct stimuli to produce specific responses to particular combinations of stimuli is of fundamental importance in neuroscience. We aim to address this question using computational models of initially non-functional, randomly connected networks of spiking neurons, which are modified by correlation-based learning rules identified experimentally. Correlation-based learning rules use the spikes of neurons to change connection strength between neurons. Correlation-based learning rules can enhance stimulus-pair representations that arise naturally in random networks. Altering the strength of inhibitory-to-excitatory connections alone was the most beneficial change, generating high stimulus-pair selectivity and reliably correct decisions across the widest range of networks. Surprisingly, changing connections between excitatory cells alone often impaired stimulus-pair selectivity, leading to unreliable decisions. However, such impairment was ameliorated or reversed by changes in the inhibitory-to-excitatory connections of those networks. Our findings demonstrate that initial heterogeneity and correlation-based changes of inhibitory synaptic strength can help generate stable network responses to stimulus-pairs.
neuroscience/behavioral neuroscience
neuroscience/neural homeostasis
neuroscience/neurodevelopment
neuroscience/neuronal and glial cell biology
computational biology
neuroscience/theoretical neuroscience
Synaptic Plasticity and Connectivity Requirements to Produce Stimulus-Pair Specific Responses in Recurrent Networks of Spiking Neurons