It has been suggested that excitatory and inhibitory inputs to cortical cells are balanced, and that this balance is important for the highly irregular firing observed in the cortex. There are two hypotheses as to the origin of this balance. One assumes that it results from a stable solution of the recurrent neuronal dynamics. This model can account for a balance of steady state excitation and inhibition without fine tuning of parameters, but not for transient inputs. The second hypothesis suggests that the feed forward excitatory and inhibitory inputs to a postsynaptic cell are already balanced. This latter hypothesis thus does account for the balance of transient inputs. However, it remains unclear what mechanism underlies the fine tuning required for balancing feed forward excitatory and inhibitory inputs. Here we investigated whether inhibitory synaptic plasticity is responsible for the balance of transient feed forward excitation and inhibition. We address this issue in the framework of a model characterizing the stochastic dynamics of temporally anti-symmetric Hebbian spike timing dependent plasticity of feed forward excitatory and inhibitory synaptic inputs to a single post-synaptic cell. Our analysis shows that inhibitory Hebbian plasticity generates ‘negative feedback’ that balances excitation and inhibition, which contrasts with the ‘positive feedback’ of excitatory Hebbian synaptic plasticity. As a result, this balance may increase the sensitivity of the learning dynamics to the correlation structure of the excitatory inputs.
There is a striking difference between the number of synaptic contacts received by a typical cortical cell, which is about 3,000 to 10,000, and the required number of about 30 excitatory inputs to bring the cell to its firing threshold [1], [2]. For example, inputs from 1,000 excitatory presynaptic cells firing at a medium rate of 10 spikes/s will yield 10010 (mean standard deviation) input spikes every 10 ms. At this high level of input, the postsynaptic cell is expected to saturate its firing rate. In addition, whereas in vitro experiments have shown that cortical cells fire relatively regularly, in vivo recordings reveal a highly irregular neural response [1]. However, fluctuations in the excitatory postsynaptic current are expected to be negligible relative to their mean, and therefore they themselves cannot account for the irregular firing of the postsynaptic cell, as observed in vivo. The prevalent explanation for these seemingly contradictory findings is that excitatory and inhibitory inputs to cortical cells are balanced; i. e. , the mean excitatory and inhibitory inputs to the cell are on the same order of magnitude. In terms of the balance hypothesis, the firing of the postsynaptic cell in the above example is not determined by the mean input of 100 excitatory postsynaptic potentials every 10 ms, which will be canceled by the mean inhibitory input, but rather by the fluctuations that are an order of magnitude smaller. Thus, the firing rate of the postsynaptic cell will not saturate and will be characterized by a high degree of variability. There is, however, some confusion as to the nature and origin of the balanced state. One approach to this problem suggested that this balance results from recurrent neural dynamics [3]–[7]. Tsodyks and Sejnowski [3] showed that feedback (i. e. , recurrent interactions) in finite chaotic networks can produce the desired balance. However, this solution requires strong interactions between the neurons and assumes a high probability of synaptic failure. Van Vreeswijk and Sompolinsky [6], [7] showed analytically that the balance of steady state excitation and inhibition can be obtained as a stable solution to the network dynamics, and requires no special fine tuning of parameters due to the feedback from the network dynamics. These studies focused on the balancing of the lateral inputs via feedback of recurrent interactions and ignored the feed forward inputs to the system. However, empirical findings have shown there is a fast balance of transient inputs to a barrel cortical cell [8], [9]. An alternative model suggested by Newsome and Shadlen [2], argues for the importance of balancing feed forward excitation and inhibition inputs in feed forward (propagating) neural networks. However, it remains unclear what mechanism underlies the fine tuning required for balancing feed forward excitatory and inhibitory inputs. In contrast to the considerable number of empirical studies on excitatory synaptic plasticity, much less is known about inhibitory plasticity. Nevertheless, evidence for STDP of inhibitory synapses is beginning to emerge [48]–[56]. Woodin et al. [49] found that in hippocampal cultures and acute hippocampal slices, inhibitory synapses are potentiated if pre- and post-spikes are paired to fire within about 20 ms of each other and are depressed when the time difference is larger than about 20 ms, irrespective of the order of firing (e. g. , similar to the STDP curve illustration in Figure 1D). In the entorhinal cortex, Haas et al. [50] reported a temporally asymmetric Hebbian STDP rule, in which an inhibitory synapse is potentiated if the inhibitory presynaptic cell fires 5–15 ms before the postsynaptic cell and is depressed if the postsynaptic cell fires 5–15 ms before the presynaptic cell, Figure 1C. Here we investigate the hypothesis that the temporally asymmetric Hebbian STDP of inhibitory synapses is responsible for the balance of transient feed forward excitation and inhibition. This issue is addressed in the framework of a stochastic dynamical model for learning feed forward inputs to a single postsynaptic cell, and then illustrated using numerical simulations or the learning dynamics of feed forward connections onto an integrate and fire neuron. We start by analyzing the stochastic learning dynamics of a single inhibitory synapse. Then we turn to investigate the learning dynamics of a population of inhibitory feed forward synapses. Finally we study a model that incorporates learning of both feed forward excitatory and inhibitory inputs to a single postsynaptic cell.
Considerable theoretical attention has been devoted to the study of learning a single excitatory synapse, see e. g. , [41], [43], and in particular the distribution of resultant synaptic weights. Two forces work to shape this distribution. The first is the positive feedback of learning an excitatory synapse (see above) that pushes the synaptic weights to their high and low saturation boundaries and hence contributes to a bimodal distribution. The second is the weight dependence of the STDP rule, which is able to decrease the strength of the positive feedback close to the saturation boundaries and hence contributes to a unimodal synaptic distribution. Following the results of Haas et al. [50] (but see also Woodin et al. [49]), we studied a family of temporally asymmetric STDP rules for the inhibitory synaptic weight of the form: (1) where is the dynamic variable that describes the synaptic strength, is the change in the synaptic strength following pre () or post () synaptic firing, is the time difference between the pre- and post-synaptic firing, is the learning rate, and and are the weight dependence and temporal filter of the STDP rule, respectively. For convenience we adopted a notation similar to that of [42]. Equation (1) defines the synaptic change due to a single pair of pre-post spikes. We shall assume that the STDP rule is additive with respect to all pairs of pre-post spike times. Equation (1) describes a temporally asymmetric STDP rule, as reported in Haas et al. [50]. The temporal filter, , can be modeled by a decaying exponent with a characteristic timescale of about 20 ms (see Figure 1A, e. g. , [11]), or by a gamma distribution, similar to the results of Haas et al. [50] (see Figure 1C). For concreteness throughout the paper in all of our numerical simulations we used with. Note that the STDP rule is temporally asymmetric and not antisymmetric due to the different scalings of depression and potentiation in, which is expressed in the dependence of the synaptic update rule. The structure of is somewhat less clear from the empirical literature; thus, for convenience of analysis, we adopted the formulation in [42], which generalizes, e. g. , [23], [41] (but see [27]): (2) (3) where is a parameter that characterizes the weight dependence of the STDP rule. Following equation (1), changes in the synaptic weight, , occur only at times where either pre or post synaptic cells have fired: (4) where is a stochastic variable which is one if the post/pre fired at time interval and zero otherwise; are the spike times of the post/pre synaptic neuron, respectively; the summation is over past times: . Note that the summation over all past spike times results from our assumption that the synaptic update rule, equation (1), is additive with respect to all pre-post spike time pairs. Taking the short time limit, , yields (5) (6) where describes the spike train of the inhibitory pre-synaptic neuron in terms of a series of delta function pulses at the spike times of the cell, (the summation is over all the spike times). Similarly describes the post synaptic spike train. We obtain: (7) (8) (9) In the limit of slow learning rate, the synaptic weight, , is relatively fixed over long periods of time, , during which the right hand side of equation (7) is sampled by the dynamics such that we can neglect its fluctuations around its mean in the limit of. This approximation yields deterministic dynamic equations for the mean synaptic weights: (10) (11) (12) where denotes averaging with respect to the distribution of the neural firing, for a given fixed synaptic weight, . To proceed with the analysis we need to specify the cross-correlation function between the pre-synaptic input and the post-synaptic response, and in particular its dependence on the synaptic weight, . However, the calculation of dependence of the temporal structure of the pre-post firing probability on the synaptic weight, even in the simple case of an integrate and fire neuron is a first-passage time problem [57], [58], which is not a trivial task. Recently, Ostojic et al [59] succeeded in analyzing the cross-correlation function between two integrate and fire neurons assuming the synaptic coupling is sufficiently weak such that the firing rate of the postsynaptic cell can be approximated by a linear function of the presynaptic input. Similarly, in this section, we assume that the synaptic coupling is sufficiently weak such that we can approximate the postsynaptic firing rate by a linear function of the presynaptic input (see also [43], [60], [61]), yielding (13) where is the pre/post synaptic mean firing rate; and the function describes the decrease in the conditional firing rate of the postsynaptic neuron at time following an inhibitory input spike at time. Substituting equation (13) into equation (10) one obtains: (14) where, and. This model highlights the following three major differences between the dynamics of inhibitory and excitatory synapses. Before deriving the full model for studying the learning dynamics of both feed-forward excitatory and inhibitory synapses, it is instructive to first study the artificial case of learning only the inhibitory inputs. We model a population of inhibitory synaptic weights, , from presynaptic inhibitory neurons projecting onto a single postsynaptic cell. Let us denote by the spike train of the th presynaptic inhibitory neuron: where are the spike times of cell (we shall omit the subscript pre hereafter). As noted above, the cross-correlation function between the pre and postsynaptic cells is an important quantity that affects the neural dynamics. In the previous section we calculated the cross-correlation function using a linear approximation to an ‘exact’ model. Instead, here and in the following section we will use an exact solution of a simplified linear model of a more abstract neuron. For analytical tractability the postsynaptic response, , is modeled to be a delayed linear sum of its inputs: ; where is small and positive to ensure causality, and represents a constant excitatory input to the cell. For simplicity we assume that the correlations are instantaneous, . We shall further assume that the statistics of the input neuron responses are isotropic; i. e. , no input neuron is statistically special. This assumption implies that: 1) The mean firing rate of all inhibitory presynaptic neurons is equal, . 2) The correlation structure of each input neuron with the rest of the input population (up to a permutation of indices) is the same. In particular, the correlation of a single input neuron with the total response of the population, , is equal for all input neurons; hence, the uniform vector is an eigenvector of the matrix. We obtain: (15) (16) (17) As above, in the limit of slow learning rate, , we can neglect the fluctuations of the synaptic weights around their mean, yielding (18) where, , and is a non-negative symmetric matrix. From the assumption of isotropy the uniform vector is an eigenvector of the correlation matrix, , with eigenvalue. From the assumption of isotropy, a homogeneous solution, , to the dynamics exists and obeys (19) with. The fixed point equation for the homogeneous solution, , is given by (20) The left hand side of equation (20), , starts from 0 at, increases monotonically in the range of, , and diverges to infinity as. The right hand side of equation (20) decreases monotonically, starts from the value of one at, crosses zero at and is continuous in the range. Hence, equation (20) has a unique solution, , in the range of, implying a net positive input to the postsynaptic cell. For the temporal derivative of the homogeneous solution, equation (19), will be positive, and for it will be negative. Hence, the uniform solution is stable to fluctuations in the uniform direction. For, equation (20) may have an additional solution with. This solution is not physical, because represents the case where the net input to the postsynaptic cell is inhibitory. A neuron with net inhibitory input will not fire and there will be no learning. To study the stability of the homogeneous solution to general perturbations, we consider an arbitrary (though small) deviation from the homogeneous solution, (note that we omitted the in the notation of the uniform solution). To first order in the deviations one obtains: (21) (22) At the homogenous fixed point, equation (20), one obtains (23) Hence, . The eigenvalues of the stability matrix obey (24) (25) where is an eigenvalue of, and is the specific eigenvalue in the uniform direction. For stability against fluctuations in the uniform direction, see above. At orthogonal directions since due to the positivity of the correlation matrix. Hence, due to the negative feedback of Hebbian learning of inhibition, the uniform solution is always stable. The STDP learning rule is an unsupervised learning rule and as such can learn salient features of the statistics of its inputs. The input statistics are expressed in the learning dynamics, equation (18), by the effective interactions between the synapses generated via the input correlations, , and the learning dynamics. Such sensitivity to input statistics may be manifested in solutions to the fixed point of the synaptic dynamics, equation (18), that reflect the correlations' structure of the input population. However, the homogenous solution, in which, always exists from the assumption of isotropy and is stable. Thus, unlike the learning dynamics of excitatory synapses, temporally asymmetric Hebbian learning stabilizes the homogeneous solution. We now turn to generalize the above formalism to study the effect of Hebbian inhibitory synaptic plasticity in the framework of a simplified model for learning excitatory synapses, , and inhibitory synapses, , constituting feed-forward input to a single postsynaptic cell. We denote by the spike train of the th excitatory/inhibitory neuron, where are the spike times of the cell. As above, the postsynaptic response, , is modeled to be a delayed linear sum of its inputs: (where is small and positive). We further assume that the system is isotropic, the mean firing rates are equal for all presynaptic neurons, (), correlations are instantaneous, , and inhibitory and excitatory inputs are uncorrelated. The excitatory synapses also follow temporally asymmetric Hebbian spike timing dependent plasticity according to equations (1) – (2) with and. In the limit of a slow learning rate the mean field synaptic dynamics are given by (27) (28) where. Note that equation (28) uses for the inhibitory synapses. The homogeneous solution, and, to the fixed point equations of the dynamics always exists, and obeys: (29) (30) where from our assumption of isotropy the uniform vector is an eigenvector of the non-negative symmetric matrix () with the corresponding eigenvalue. Fluctuation analysis around the homogeneous fixed point yields: (31) (32) (33) Let us denote by the eigenvectors of () with the corresponding non negative eigenvalue; note that is the uniform dimensional vector. The eigenvectors of the full stability matrix, , which is given by equation, are of the form, and, for. Using the fixed point equations (29) – (30) one can show that the homogeneous solution is always stable to fluctuations in the homogeneous direction. Additionally, similar to the analysis of the previous section, the homogeneous solution is always stable to fluctuations in directions of modifying the inhibition, . However, the homogeneous solution is not always stable with respect to fluctuations in non-homogeneous directions of the excitation. This point has been discussed at length in [28], [29], [42]. Essentially, as the positive feedback of the STDP dynamics of the excitatory synapses becomes strong (i. e. , for small) the homogeneous solution of the excitatory synapses loses its stability and the learning dynamics becomes more sensitive to the correlation structure of its excitatory inputs. Specifically, the eigenvalues of the stability matrix in the directions of non-homogeneous fluctuations of the excitatory synapses, (), are: . The term has to stabilize non-homogeneous fluctuations in the th excitatory direction, . For small and sufficiently large the homogenous solution will lose its stability. In addition, note that here is proportional to the deviation from the balance, which is governed by the correlations between the inhibitory inputs. As the correlations between the inhibitory neurons decrease the net input to the postsynaptic cell becomes more balanced and thus the homogeneous solution becomes less stable. Nevertheless, since the learning dynamics of the inhibitory neurons, equation (28), is only sensitive to the mean excitatory input, , and the directions of instability of the homogeneous solution are only in a heterogenous direction of the excitatory synapses and not the inhibitory, inhibition is still expected to remain uniform, obeying: (34) Hence, importantly, we find from equation (34) that the negative feedback of inhibitory plasticity works to balance the net excitatory input to the cell. In the absence of cross-correlations between inhibitory synapses, , our model predicts that Hebbian STDP dynamics converge to a complete balance of excitation and inhibition, in the limit of large. In the presence of correlations, the net inhibitory input will scale linearly with the net excitatory input (for); however, with a coefficient that is less than one. In this case; i. e. , in the presence of correlations, the deviation from exact balance will scale with the magnitude of the fluctuations. Thus, the fluctuations in the feed forward synaptic inputs to the cell will not be negligible relative to their mean.
We have studied the computational effect of temporally asymmetric Hebbian plasticity of feed forward inhibition. Hebbian plasticity of inhibition generates negative feedback, in contrast to the positive feedback generated by Hebbian plasticity of excitation. This can be understood by the following intuitive explanation. If the feed forward inhibitory synapse is very strong, then it is less likely that a postsynaptic spike will follow a presynaptic spike. As a result more pre-post spike pairs will fall on the acausal branch of the STDP learning curve than on the causal branch. This, in turn, will depress the strong synapse. On the other hand, if the synapse is weak, then pre and post spike times will be largely uncorrelated and the STDP dynamics will sample uniformly both branches of the STDP curve with equal probability. If the area under the acausal branch is smaller than the area under the causal branch, (for), the weak synapse will potentiate (the case of is not interesting since it reduces all inhibitory synapses to zero). This negative feedback, in the case of temporally asymmetric Hebbian plasticity and instantaneous correlations as was studied here, implies that the inhibitory synaptic weight distribution is unimodal and that the homogeneous solution of learning a population of inhibitory synapses is stable, even when the homogenous solution for the excitatory synapses loses its stability. However, in our analysis we focused on a simple case where there are no correlations between excitatory and inhibitory neurons. Incorporating such correlations to our model adds a term of the form to the right hand side of equation (28). This term will cause the homogeneous solution for the inhibition to cease to exist when the uniform solution to the excitation loses its stability. Yet the increased stability of the homogeneous solution of the inhibition suggests that the inhibitory feed forward input to a cell is expected to be more broadly tuned than the excitatory input. In addition we found that inhibitory Hebbian plasticity works to balance the net excitatory inputs of the cell. Two terms govern the homogeneous fixed point of the inhibitory synapses. The first term results from the contribution of the product of the mean firing rates to the pre-post full-correlations (this is a generalization of the fixed point equation of equation (28) to the case): , where. The second term results from the contribution of covariation in the firing: . The first term, works to balance the net inhibitory and excitatory inputs to the post-synaptic cell. In the absence of correlations between the pre-synaptic inhibitory neurons, , the contribution of the covariance term will decay as. In this case, neglecting the covariance term, the inhibitory fixed point will balance the net excitatory input: , as long as the net excitatory input is not too large: . Thus, in the absence of correlations, Hebbian STDP of inhibition will balance the excitation, even for the case. In the presence of correlations, for the case, inhibition will balance excitation in the sense that it will scale linearly with it, this balance is skewed towards excitation, as was reported, e. g. , in Heiss et al. [9]. For, inhibition will not scale linearly with excitation. However, the deviation from exact balance (i. e. ,) is expected to scale (not necessarily linearly) with the magnitude of the fluctuations, , for both the and cases. The balance has a twofold effect. First, balancing the mean excitation and inhibition inputs to the cell increases the relative contribution of the fluctuations to the cell' s response. Note that we find that even if exact balance is not obtained the fluctuations are not expected to be negligible relative to the mean input. Second, the exact balance reduces the stability of the homogeneous solution to the learning dynamics to fluctuations of excitatory synaptic weights in a non-homogeneous direction. Hence, it increases the sensitivity of the STDP dynamics to the structure of the excitatory input. Our analysis was performed using the framework of simplified models of postsynaptic neural response. Although these simplified models provide a mere caricature of the rich response dynamics of a neuron to its synaptic inputs, they further our analytical understanding of the possible outcomes of Hebbian inhibitory synaptic plasticity, which cannot be achieved if the neural response dynamics need to be solved numerically. Furthermore, numerical simulations support our analytical results qualitatively beyond the framework of the simplified linear neuron model. We introduced several additional simplifying assumptions to our model. First, we focused on temporally asymmetric Hebbian STDP for the inhibitory synapses. However, STDP rules may be highly variable and this variability may be manifested by a qualitative difference from our results. For example an anti-Hebbian STDP rule, such as depicted in Figure 1B, may generate positive feedback to the inhibitory plasticity, instead of the negative feedback reported here. Woodin et al. [49] reported a temporally symmetric STDP rule in which the learning rule acts as a coincidence detector of post- and pre-spikes, Figure 1D. This learning rule may act to shape the timing of postsynaptic cell spikes by ‘selecting’ the inhibitory inputs that fired in a specific time interval around the excitatory input. Such a mechanism would require a specific temporal correlations structure between the input inhibition and excitation. Of particular interest is the ‘thalamocortical’ circuit, in which feed forward excitatory inputs arrive directly from thalamus to cortex whereas inhibitory inputs result from a relay of the thalamic input via local (cortical) interneurons. This network architecture manifests in a rich temporal and functional (similarity of preferred stimuli) correlation structure between excitation and inhibition, which may have a significant qualitative effect on learning dynamics. However, here we assumed a very basic correlations structure between the responses of the presynaptic neurons. The investigation of the different effects of various STDP rules and the elaborate spatial and temporal correlations structure of presynaptic neuron responses is beyond the scope of the current paper. Nevertheless, this work suggests a theoretical framework for addressing these issues.
One of the longstanding enigmas in neuroscience is the origin of inherent neural noise. It has been suggested that this noise results from a careful balance of excitatory and inhibitory inputs to neurons. Obtaining this balance requires fine tuning of the relative strengths of the inhibitory and excitatory inputs to each cell. However the mechanism that enables this fine tuning of parameters remains unclear. We suggest that a balance of excitatory and inhibitory inputs can be achieved via a process of unsupervised learning of the inhibitory synaptic strengths. We find that whereas Hebbian learning induces strong positive feedback on excitatory inputs that acts as a force that pulls them towards their boundaries, Hebbian learning of inhibition induces negative feedback. This negative feedback acts to balance the average excitatory input to the cell and makes the learning process less sensitive to the statistics of the inhibitory inputs. Surprisingly, this balance increases the sensitivity of learning to the statistics of the excitatory inputs. Thus, the balance of feed-forward excitation and inhibition emerges as a natural outcome of Hebbian learning applied to the inhibitory inputs to the cell.
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neuroscience
Balancing Feed-Forward Excitation and Inhibition via Hebbian Inhibitory Synaptic Plasticity