Hebbian plasticity describes a basic mechanism for synaptic plasticity whereby synaptic weights evolve depending on the relative timing of paired activity of the pre- and postsynaptic neurons. Spike-timing-dependent plasticity (STDP) constitutes a central experimental and theoretical synaptic Hebbian learning rule. Various mechanisms, mostly calcium-based, account for the induction and maintenance of STDP. Classically STDP is assumed to gradually emerge in a monotonic way as the number of pairings increases. However, non-monotonic STDP accounting for fast associative learning led us to challenge this monotonicity hypothesis and explore how the existence of multiple plasticity pathways affects the dynamical establishment of plasticity. To account for distinct forms of STDP emerging from increasing numbers of pairings and the variety of signaling pathways involved, we developed a general class of simple mathematical models of plasticity based on calcium transients and accommodating various calcium-based plasticity mechanisms. These mechanisms can either compete or cooperate for the establishment of long-term potentiation (LTP) and depression (LTD), that emerge depending on past calcium activity. Our model reproduces accurately the striatal STDP that involves endocannabinoid and NMDAR signaling pathways. Moreover, we predict how stimulus frequency alters plasticity, and how triplet rules are affected by the number of pairings. We further investigate the general model with an arbitrary number of pathways and show that depending on those pathways and their properties, a variety of plasticities may emerge upon variation of the number and/or the frequency of pairings, even when the outcome after large numbers of pairings is identical. These findings, built upon a biologically realistic example and generalized to other applications, argue that in order to fully describe synaptic plasticity it is not sufficient to record STDP curves at fixed pairing numbers and frequencies. In fact, considering the whole spectrum of activity-dependent parameters could have a great impact on the description of plasticity, and a better understanding of the engram.
Synaptic plasticity, one of the paramount biological mechanism supporting learning and memory in the brain, has been the object of a wide literature spanning from experimental works [1–3] to computational investigations [4–6]. In 1949, Donald Hebb’s pioneering work postulated that long-term modifications of the synaptic efficacy can be induced in response to patterns of activity of the pre- and postsynaptic neurons [7]. Since then, many experimental studies have confirmed and extended Hebb’s postulate and have highlighted the complexity of the signaling pathways and their neuromodulation leading to synaptic efficacy changes in response to pre- and postsynaptic activity patterns [1,2, 8,9]. Numerous mathematical models were also developed to emulate this diversity and infer their computational capabilities [4–6]. Spike-timing-dependent plasticity (STDP) is a biological phenomenon of activity-dependent change in synaptic connectivity that is viewed as a synaptic Hebbian learning rule. STDP has been widely studied in the last two decades and experimentally observed at many synapses in various forms, and those were classified depending on the manner in which they implement Hebb’s postulate [8,9]. STDP is assessed experimentally through repetitive paired activations of the pre- and postsynaptic sites with a prescribed timing that is denoted in this paper Δt. By convention, we consider Δt < 0 when the postsynaptic stimulation occurs before the paired presynaptic one (post-pre pairings), and Δt > 0 when the presynaptic stimulation occurs before the postsynaptic one (pre-post pairings). Classically, the same paired stimulation is repeated between 80 and 150 times at a constant frequency (between 0. 1 and 5 Hz) [8–10]. In many cases, these pairing patterns induce long-term plasticity exhibiting various polarities (increase or decrease of the synaptic weight as a function of the sign of Δt) or magnitudes. In the vast majority, the expression of STDP is restricted to a narrow interval of values for Δt; thus, when pre- and postsynaptic activities are separated by a large Δt (|Δt| > 50 ms in most of the cases), no long-term plasticity is observed [11,12]. In this study, we term Hebbian STDP the plasticities whereby sequences of presentations of a presynaptic spike followed by a postsynaptic spike lead to Long-Term Potentiation (LTP) when repeated a specific number of times (denoted NPairings) at a certain frequency, whereas reverse sequences induce Long-Term Depression (LTD). Hebbian STDP was reported in various structures such as the hippocampus [11,13–15], the cerebral cortex [12,16–19] and the striatum [20–23]. Conversely, we will term here (bidirectional) anti-Hebbian STDP, the forms of STDP exhibiting a reverse polarity when compared to the aforementioned Hebbian STDP: causal pre-post pairings induce LTD and anti-causal post-pre pairings induce LTP. Bidirectional anti-Hebbian STDP was also observed, for instance in the striatum [24–28], in the somatosensory cortex [29] or in the cerebellum-like structure of the electrical fish [30]. Unidirectional anti-Hebbian STDP, i. e. LTD for both post-pre and pre-post pairings, is another main form of STDP observed in the neocortex [31,32], the dorsal cochlear nucleus [33], the cerebellum [34,35] and the hippocampus [36]. We underline that different definitions of (anti-) Hebbian STDP were used in the literature; the present study follows the terminology of early experimental studies [11,12], or Figure 2 of the review [8], but differs, e. g. , from the definitions used in [37]. These plasticities were shown to be dependent upon the parameters of the stimulation beyond spike-timing: for instance, varying the frequency at which pairings are presented or the total number of pairings, presenting distinct spike patterns (triplets, single spike, theta bursts, …) [17,38–41] or changing neuromodulatory tones [21] may lead to distinct forms of STDP. Despite the existence of multiples forms of STDP [8,9], all of them have in common the crucial role played by the calcium transients in the pre- and postsynaptic compartments for the induction and maintenance of plasticity. Postsynaptic calcium influxes through NDMA receptors (NMDAR) and voltage-sensitive calcium channels have been demonstrated to be key factors governing plasticity expression and polarity [10]. Regarding Hebbian plasticity, calcium-dependent mechanisms act as coincidence detectors, essential to implement any type of STDP. In addition, distinct signaling pathways appear to be involved, namely (i) calcium triggering downstream cascades modulating calcium/calmodulin-dependent kinase II (CaMKII) [42] which ultimately regulates the gene expression and/or (ii) the endocannabinoid (eCB) system, whose synthesis and release is calcium-dependent, acting retrogradely on the presynaptic element [43–45]. Importantly, both of these pathways are able to trigger LTP or LTD depending on the spatio-temporal kinetics of the calcium [19,40,46]. Calcium dynamics thus constitute a key factor in synaptic plasticity induction and in selecting plasticity forms. Accordingly, numerous mathematical models were based on calcium transients and described various forms of STDP [5]. In particular, Graupner and Brunel [47] proposed simple calcium-based models able to account for a wide range of experimental observations on synaptic plasticity. However, while calcium-based models succeed in reproducing the results of the “classical” STDP (∼ 100 pairings), they do not take into account the dynamics of the establishment of plasticity and the variety of timescales involved in plasticity induction. Indeed, in computational neuroscience, it is implicitly admitted that the synapse gradually amplifies synaptic changes as the number of stimulus presentation increases to reach the final plasticity profiles. However, plasticity occurs at vastly distinct timescales and protocols based on one hundred trials (i. e. , pairings), as classically performed in STDP experiments, only reveal an extreme steady state outcome. Actually, a dozen of trials can be sufficient to induce plasticity, if not less in the case of associative memory [39,40,48,49]. Importantly, it was recently shown that depending on the number of pairings, STDP on the cortex-to-striatum synapses (cortico-striatal STDP) exhibits three distinct forms of plasticity: an NMDAR-mediated LTP and an eCB-mediated LTD for 100 post-pre and pre-post pairings, respectively [20,21,24,25,27], and an eCB-mediated LTP for 5-15 post-pre pairings [39,40]. Note that at cortico-striatal synapses, GABA operates as a Hebbian/anti-Hebbian switch [27,28] and that without the blockade of GABA transmission, an anti-Hebbian STDP is induced as observed in vivo [26]. These phenomena were reproduced in a biophysical model of the cortico-striatal synapse accounting for receptors activation dynamics (36 equations and 150 parameters) [40]. However, no simple phenomenological model reproduces these phenomena, and in particular models of plasticity based on the calcium hypothesis fail to reproduce such complex dynamical emergence of plasticity. Here, we propose a new model built upon the calcium hypothesis and taking into account the existence of multiple signaling pathways at a given synapse that may be activated at distinct calcium levels. We instantiate one model to fit the data from cortico-striatal STDP [39,40], and show that the model accurately reproduces the experiments on the dependence of STDP on both the number and frequency of pairings. We use this model to predict the response of the system as the stimulus frequency and number of presentations are varied, and extend the model to show how triplet rules will depend on the number of stimulus presentations. Our model goes beyond the case of the cortico-striatal synapse with two signaling pathways, and we further explore the diversity and limited range of dynamical plasticity establishments that can be unfolded from classical Hebbian STDP. In the face of this diversity, we further propose experimentally implementable protocols to differentiate those scenarii. This study thus sheds a new light on the interplay of multiple signaling pathways at single synapses and how this multiplicity endows the synapse with the capacity of encoding multiple STDP profiles depending on the number and frequency of stimulus presentation, and argues that experiments based on stereotypical stimulus presentations are not sufficient to finely account for the complexity of plasticity, even in widely studied synapses.
We developed a general calcium-based model of the synapse allowing to take into account the presence of multiple pathways and past activity in the establishment of plasticity. Our developments build upon the Graupner and Brunel model [47], and extend it by (i) introducing multiple plasticity pathways, and (ii) taking into account the fact that receptor activation thresholds may depend on past activity. We provide here the details of the models and the emergent changes in synaptic weight, as well as a theoretical formula thereof. The model we built in the previous section is general and is thus able to reproduce a variety of synapses and plasticity mechanisms relying on calcium dynamics. We study in this section the case of STDP at the cortico-striatal synapse, which was studied experimentally with varying Npairings [39,40]. In these contributions, it was shown that STDP at the cortico-striatal synapse relied both on NMDAR and endocannabinoid pathways (see Fig 1 (a) ), and that synaptic changes after paired pre- and postsynaptic spikes not only depended on the timing between the pre- and postsynaptic spikes, but also varied with the number and the frequency of the pairings presented. Namely, it was shown that for pre-post pairings (0 < Δt < +40 ms), an eCB-LTD progressively appeared as the number of pairings was increased, while for post-pre pairings (−30 ms < Δt < 0 ms), a biphasic STDP emerged with an eCB-LTP for a low number of pairings (5 − 15 pairings), an absence of plasticity between 25 and 50 pairings, leaving room for NMDAR-LTP at higher numbers of pairings (≥ 75 pairings). A schematic representation of the biological pathways involved is provided in Fig 1 (a) together with the biophysical mechanisms and proteins cascades (described in more detail in [40]). A minimal model of cortico-striatal plasticity thus requires taking into account two different and independent calcium-dependent pathways (P = 2), an eCB-dependent mechanism (α = e) which induces both LTP or LTD depending on the specific timing Δt of the pairings, and an NMDAR-dependent (α = n) associated to LTP only. This yields to the following system of stochastic differential equations (see schematic diagram in Fig 1 (b) ): { τdρedt=−ρe (1−ρe) (ρ*−ρe) +γep (1−ρe) Θ[ c (t) −θep (c˜t) ]−γedρeΘ[ c (t) −θed (c˜t) ]+Noisee (t) τdρndt=−ρn (1−ρn) (ρ*−ρn) +γnp (1−ρn) Θ[ c (t) −θnp (c˜t) ]+Noisen (t) (6) The complete synapse model is made of N ≫ 1 independent pairs (ρ e i, ρ n i) i = 1 ⋯ N satisfying Eq (6), and a synaptic change deduced from the proportion of synapses that switch from being potentiated to depressed or reciprocally, through the sigmoidal map H of Eq (4). As described above, in response to pre- and postsynaptic spike-timing (Δt = tpost − tpre), the calcium dynamics c undergoes jumps followed by exponential relaxation as described in Eq (2), activating eCB-LTP, eCB-LTD and NMDAR-LTP as soon as c exceeds specific LTP or LTD thresholds (see Fig 1 (c), where the LTP and LTD thresholds are represented by the green and red lines respectively) with both thresholds taken from the adjusted parameters of Table 1 for the eCB pathway. When only one presynaptic spike (thus without postsynaptic spike) is evoked, the calcium concentration amplitude exceeds the LTP threshold for a short amount of time, and remains below the level of LTD induction: repeating this protocol does not lead to significant plasticity. For a pre-post stimulation (Δt > 0), the summation of the pre- and postsynaptic calcium spikes triggers both LTP and LTD. The same is valid for Δt < 0, but the relative time spent above the LTP and LTD thresholds would be significantly different depending on Δt, underlining the importance of the timing and order between the spikes in the resulting plasticity: in the example depicted in Fig 1 (c) and parameters γ α x from Table 1, a pre-post stimulation yields LTD whereas a post-pre stimulation yields LTP, consistent with anti-Hebbian STDP at cortico-striatal synapses ex vivo [27,28] in the absence of GABAA receptor antagonist and in vivo [26]. STDP at the cortico-striatal synapse, studied in the previous section, provides a realistic example of plasticity with multiple pathways. Our model, relying on only two equations and a small number of biologically interpretable parameters emulating NMDAR- and eCB-dependent pathways, reproduces all the phenomena reported at the cortico-striatal synapse, and allowed to draw predictions on plasticity for more complex stimuli such as triplet rules. The present model is however much more general than the case of the cortico-striatal synapse: it can indeed emulate synapses with more than two signaling pathways with arbitrary independent plasticity rules, and thus allows unraveling the dynamics of plasticity expression in a variety of synapses with distinct plasticity. Interestingly, while being quite versatile, the repertoire of behaviors that can be reproduced given a fixed number of pathways remains limited, and the model thus also provides predictions on the minimal number of pathways involved given a plasticity profile. Indeed, a single pathway shall induce a monotonic establishment of plasticity if there is no inactivation of the pathway, whereas situations with two pathways can lead to four changes of plasticity (LTP and LTD inactivation for each of the two pathways), possibly with periods of non-significant synaptic changes. More generally, plasticity with P pathways may lead to up to 2P changes of monotonicity, possibly interspersed with periods of non-significant plasticity. We investigate in the next sections a few possible scenarii relying on at most two signaling pathways that could lead to Hebbian or anti-Hebbian plasticity and suggest experiments that could distinguish distinct situations.
Synaptic plasticity is a complex phenomenon relying on the activation of a number of receptors and signaling pathways [3,10]. A substantial difficulty for experimentalists is to characterize plasticity in the large variety of possible situations occurring in vivo. To reduce this complexity, a protocol designed to reveal plasticity consists in considering changes in synaptic transmission after the reiterated presentation of a fixed spike pattern a large number of times (on the order of one hundred) and at a slow rate. From these experiments, it remains complex to decipher the multiple signaling pathways involved in the expression of plasticity, and their complex interplay, particularly for low numbers of stimulus presentations or for various pairing frequencies. To disentangle the distinctive role of multiple pathways, we developed and studied a phenomenological model of the evolution of synaptic weights and tested its responses in distinct situations. The model relies on calcium transients triggered by the spiking activity of neurons on both sides of the synapses, and is built upon previous theoretical works (see [47] and references therein). When plasticity (LTP and LTD) relies on multiple signaling pathways [10], the timescales at which these mechanisms activate and inactivate upon repetitive stimulation can lead to a variety of behaviors as a function of the number and of the frequency of pairings, which cannot be inferred from experiments where those are fixed. Our model proposes a general and minimal framework to integrate multiple signaling pathways and their dependences upon repetitive stimulations. We have instantiated this model with two specific pathways, NMDAR- and eCB-dependent, that was inspired by experiments at cortico-striatal synapse showing variations of the emergent plasticity upon variation of the number of pairings [39,40]. Our model reduces to two stochastic equations Eq (6) and a small number of parameters, and accurately reproduced the data obtained in that experimental contribution. To our knowledge, this model is the most parsimonious model reproducing STDP experimental results, yet many models of the class that we introduced can be proposed, including for instance NMDAR-LTD or pathways activated by distinct molecules. We also used the model to predict the response of the system when the number of stimulations, the pairing frequency or the number of spikes, are varied. This led us to draw predictions on the modifications of STDP profiles when the frequency of stimulus presentations was varied. Eventually, we have made new predictions on the dependence of triplet rules upon the number of stimulus presentations, and showed that complex non-monotonic STDP profiles emerge with up to three distinct phases. Our model goes beyond the particular case of the cortico-striatal synapse for which data was available, and we pursued our investigations by considering distinct mechanisms that could underlie another type of plasticity, symmetric anti-Hebbian LTD (with LTD for pre-post and post-pre pairings). In this case, we investigated three distinct possible scenarii involving up to two distinct pathways, and showed that unexpected phenomena may arise upon variations of the number and frequency of pairings, and in particular the emergence of an LTP at 100 pairings for high frequencies. Overall, these results highlight the fact that electrophysiological experiments at a fixed frequency and a prescribed number of pairings may not be sufficient to extrapolate to other situations with smaller numbers of pairings or presentation frequencies. To our knowledge, the present model is the first to take into account distinct signaling mechanisms involved in plasticity in a simple and compact framework. The simplicity of the present model allows to envision the implementation of this type of synapse at the level of a neural network, opening the way to theoretical studies of information processing capacity of networks endowed with complex activity-dependent plasticity rules. In addition to the development of a framework integrating multiple pathways, one of the main novelties of this model compared to other calcium-based models is that we have explicitly incorporated activity-dependent thresholds allowing to recover the response of plasticity mechanisms on the past activity of cells. In the present model, we simply assumed that this history-dependence is parameterized by a cumulative calcium concentration. Explicitly incorporating this dependence allows taking into account in the model multifarious experimental facts including finiteness of the calcium pool in the postsynaptic compartment, desensitization of synaptic-receptors and homeostasic mechanisms [63]. The present model proposing that this dependence on past activity relies on cumulative calcium constitutes a first step, and could be refined in several directions, for instance incorporating a slow decay of past-activity dependence with time (considering integrated calcium spikes with an exponentially decaying kernel for instance), moving averages in the flavor of sliding thresholds in the classical Bienenstock-Cooper-Munro (BCM) rule [56,57]. In our case, the average activity of the neuron would be simply modeled by postsynaptic calcium concentration (a reasonable proxy of neural activity), or with more refined models involving distinct molecular species and their timescales. Despite a good qualitative agreement and an improved accuracy on the dynamics of the expression of plasticity, we found that our model shows a slight mismatch in the timescales at which plasticity emerges: first, although experiments at cortico-striatal synapses show a significant plasticity arising as early as 5 pairings and reaching a maximum at 10 pairings, we did not find in the model significant plasticity at 5 pairings and the maximal plasticity occurred after a slightly larger number of 12 pairings. Moreover, a unidirectional LTP in the cortico-striatal plasticity at 100 pairings was observed experimentally when the frequency of pairing presentations reached 4 Hz, while the model reproduces this phenomenon slightly above 30 Hz. We believe that this slower response of the present model relies on the bistable nature of the model, following [47]. This bistability makes the system quite rigid and resistant to rapid changes, and a direct perspective would be to implement a more flexible model dropping the bistable model but conserving the long-term stability of macroscopic synaptic strength ensured by the bistable potential. The present model would be also used in future works focusing on the implementation of the cortico-striatal STDP in large stochastic neural networks, with several classes of interneurons, aimed at understanding the possible role of implementing distinct cortico-striatal plasticity, in particular LTP, arising at various timescales and their possible role in information processing in striatum. All in all, the present model suggests to reconsider a current widely admitted implicit hypothesis in models, and questions the usual view of STDP in models that consider a fixed curve solely dependent on the spike timing (Δt). Indeed, in most neural network models with STDP, it is considered that synaptic coefficients are progressively incremented depending on spike timings and according to toy-models of STDP (e. g. , double-exponential curves). This is implemented in various manners, including additive or multiplicative changes depending on all spike pairings or on the nearest-spike (see e. g. [64]). At the level of networks, a number of stochastic models were developed to study the influence of STDP as a synaptic plasticity rule (see the review [4]). In particular, early works showed the influence of classical Hebbian and asymmetric STDP in the dynamics of neuronal networks [65–67]. The role of STDP-based rules in the emergence of structures in recurrent neural networks was also studied in a series of papers, highlighting for instance a possible key impact on the self-organization of microcircuits [68–70]. More recently, the distribution of synaptic weights and its stability in randomly stimulated networks with different triplet rules has been extensively studied [37]. The activity-dependent rule we proposed here, reproducing variable synaptic changes as a function of the number of stimulations, may lead to significant changes in the resulting connectivity and dynamics of neural networks. Our model offers an avenue to revaluate the possible modifications of the resulting dynamics emphasizing the role of timescales in these systems [71].
Simulations were performed with a custom code implemented in Python 2. 7 or 3. 5, developed within the Spyder environment of the Anaconda suite (Anaconda Software Distribution. Computer software. Vers. 2-2. 4. 0. Continuum Analytics, Nov. 2015. Web. <https: //continuum. io>). The main modules used numpy, matplotlib, math, scipy. Elementary simulations of the model were run on a Macbook Pro (Intel Core i5 processor and 16 RAM) and more demanding simulations were executed on the Inria Paris-Rocquencourt computer cluster. Figures and plots were realized using matplotlib module of Python and Illustrator/Photoshop of the Adobe series. Unless specified otherwise, we used the parameter values listed in Table 1. These parameters were optimized starting from initial guesses chosen for consistency of the model with the data using the extensive analysis of [47, Fig 2]. For adjusting our parameters to the cortico-striatal plasticity, we used a global optimization algorithm, the differential-evolution function from scipy. optimize module to obtain qualitative fits. Simulations of the model were realized either from the theoretical expressions computed, or with numerical simulations of the system of stochastic equations Eq (6). We used temporal discretization using an Euler scheme on t = −1…101 s with Niter steps (see Table 1) and run the simulation for a set of N = 1000 individual efficacies. To compute the change in macroscopic synaptic strength for the different pairings, we run the simulation for NPairings = 100 pairings and store the results for all the pairings during the STDP protocol. Therefore, for each fixed Δt, the results obtained for different pairings are not independent, which has the interest of uncovering the evolution of one given synapse, and has no impact on the global outcome of the simulations as can be seen when compared with analytical results. Except for Fig 3 where the analytical mean-field solutions are represented, all the figures show the numerical simulations. For Fig 6, we have reproduced 30 independent simulations in parallel to obtain the statistical means and standard deviations depicted. All heatmaps used a logarithmic color bar to represent changes in synaptic strength. The classification in mono-, bi- or tri-phasic regimes in Fig 7 (c) was done through a visual inspection of the STDP curves associated to each of the 20 Δt1 and Δt2. The model we studied is nonlinear, and as such, it was complex to derive the explicit form of the probability distribution of the solutions. Following the approach proposed in the Appendix of [47], we derived the probability distribution of the solution of an approximate model valid when the system remains in the linear part of the cubic bistable term. The model involves a linear Ornstein-Uhlenbeck with deterministic time-dependent coefficients α (t) and β (t) that we computed as follow. The solution of linear stochastic differential equations of type [72] (with B a standard Brownian motion): d ρ (t) = (α (t) ρ (t) + β (t) ) d t + σ (t) d B (t) (7) with initial condition ρ0 can be easily expressed in closed form as: ρ (t) = ρ 0 exp (∫ 0 t α (s) d s) + ∫ 0 t exp (∫ u t α (s) d s) β (u) d u + ∫ 0 t exp (∫ u t α (s) d s) σ (u) d B (u). (8) As indicated in the main text, the synaptic change is obtained using a sigmoid transform of the proportion U of synapses that, after the protocol, have crossed upwards the threshold value ρ*, over the proportion D of those that crossed downwards. Since the above described Ornstein-Uhlenbeck process is Gaussian, these probabilities are fully characterized by the mean and single-time variance functions of ρ, which have the following expressions: E [ ρ (t) | ρ 0 ] = ρ 0 exp (∫ 0 t α (s) d s) + ∫ 0 t exp (∫ u t α (s) d s) β (u) d u (9) and Var [ ρ (t) | ρ 0 ] = ∫ 0 t exp (2 ∫ u t α (s) d s) σ 2 (u) d u (10) We thus derive the time-varying coefficients α and β arising in the approximated model, for the eCB pathway (the other can be dealt with in the same way). These are computed describing the time spent above the various thresholds of the model. We denote by η i x the average time spent above the threshold θ i x: this quantity only depends on the calcium dynamics can be easily computed analytically for each given a pairing protocol. Similarly, we define te = T ne the time at which the eCB-LTP is inactivated at the cortico-striatal synapse, with T being the duration between two pairings and ne the pairing number at which eCB-LTP is first inactivated. Denoting Γ i x = γ i x η i x, we have the following compact formulae for the coefficients of the Ornstein-Uhlenbeck processes αe, βe and σe: α e (t) = { - Γ e d + Γ e p τ = - 1 τ 1 if t < t e - Γ e d τ = - 1 τ 2 if t e < t (11) β e (t) = { Γ e p τ = ρ ˜ 1 τ 1 si t < t e 0 si t e < t (12) σ e (t) = { η e d + η e p τ σ = σ 1 τ 1 si t < t e η e d τ σ = σ 2 τ 2 si t e < t (13) Because of the simple, piecewise constant form of the coefficients, we have, for deterministic initial conditions: E [ ρ e (t) | ρ e (0) ] = { ρ e (0) exp (- t τ 1) + ρ ˜ 1 (1 - exp (- t τ 1) ) if t < t e ρ e (0) exp (- t - t e τ 2 - t e τ 1) + ρ ˜ 1 exp (- t - t e τ 2) (1 - exp (- t e τ 1) ) if t e < t (14) and Var [ ρ e ] = { σ 1 2 (1 - exp (- 2 t τ 1) ) t < t e σ 1 2 exp (- 2 t - t e τ 2) (1 - exp (- 2 t e τ 1) ) + σ 2 2 (1 - exp (- 2 t - t e τ 2) ) if t e < t (15) The probability that an initially depressed synapse becomes potentiated is thus given by: U e (n T) = P (ρ e > ρ * | ρ e (0) = 0) = 1 2 (1 + erf (− ρ * − E [ ρ e | ρ e (0) = 0 ] (n T) Var [ ρ e ] (n T) ) ), (16) and the probability of an initially potentiated synapse to become depressed by: D e (n T) = P (ρ e < ρ * | ρ e (0) = 1) = 1 2 (1 − erf (− ρ * − E [ ρ e | ρ e (0) = 1 ] (n T) Var [ ρ e ] (n T) ) ). (17) allowing directly to obtain the change in synaptic weight associated as H (U e (n T) D e (n T) ). A comparison of the Ornstein-Uhlenbeck approximation with the numerical simulations of the nonlinear system is provided in Fig 3 (a) and S2 (a) Fig, showing a good agreement for the parameter set chosen. The data used to fit and validate our results were previously published in [39,40]. We refer to these papers for more specific information on the experimental protocol.
The brain’s capacity to treat information, learn and store memory relies on synaptic connectivity patterns, which are altered through synaptic plasticity mechanisms. Experimentally, such plasticities were evidenced through protocols involving numerous repetitive stimulations of a given synapse, and were shown to be supported by multiple pathways. Using a simple biologically grounded mathematical model, we show how activation timescales and inactivation levels of each pathway interact and alter plasticity in an intricate manner as stimuli are presented. Building upon data from the synapse between cortex and striatum, we show that synaptic changes may revert or re-emerge as stimuli are presented, and predict specific responses to changes in stimulus frequency or to distinct simulation patterns. Our general model shows that a given plasticity profile emerging in response to a repetitive stimulation protocol can unfold into various scenarii upon variations of the number of stimulus presentations or patterns, which tightly depends on the underlying activated pathways. Altogether, these results argue that in order to better understand learning and memory, single plasticity responses obtained through intensive stimulations do not reveal the complexity of the responses for smaller number of presentations, which may have a strong impact in fast learning of stimuli with low numbers of presentations.
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Interplay of multiple pathways and activity-dependent rules in STDP