Emergent Dynamical Properties of the BCM Learning Rule
 Lawrence C. Udeigwe^{1},
 Paul W. Munro^{2} and
 G. Bard Ermentrout^{3}Email author
DOI: 10.1186/s1340801700446
© The Author(s) 2017
Received: 15 September 2016
Accepted: 18 January 2017
Published: 20 February 2017
Abstract
The Bienenstock–Cooper–Munro (BCM) learning rule provides a simple setup for synaptic modification that combines a Hebbian product rule with a homeostatic mechanism that keeps the weights bounded. The homeostatic part of the learning rule depends on the time average of the postsynaptic activity and provides a sliding threshold that distinguishes between increasing or decreasing weights. There are, thus, two essential time scales in the BCM rule: a homeostatic time scale, and a synaptic modification time scale. When the dynamics of the stimulus is rapid enough, it is possible to reduce the BCM rule to a simple averaged set of differential equations. In previous analyses of this model, the time scale of the sliding threshold is usually faster than that of the synaptic modification. In this paper, we study the dynamical properties of these averaged equations when the homeostatic time scale is close to the synaptic modification time scale. We show that instabilities arise leading to oscillations and in some cases chaos and other complex dynamics. We consider three cases: one neuron with two weights and two stimuli, one neuron with two weights and three stimuli, and finally a weakly interacting network of neurons.
Keywords
BCM Learning rule Oscillation Chaos1 Introduction
For several decades now, the topic of synaptic plasticity has remained relevant. A pioneering theory on this topic is the Hebbian theory of synaptic modification [1, 2], in which Donald Hebb proposed that when neuron A repeatedly participates in firing neuron B, the strength of the action of A onto B increases. This implies that changes in synaptic strengths in a neural network is a function of the pre and postsynaptic neural activities. A few decades later, Nass and Cooper [3] developed a Hebbian synaptic modification theory for the synapses of the visual cortex, which was later extended to a threshold dependent setup by Cooper et al. [4]. In this setup, the sign of a weight modification is based on whether the postsynaptic response is below or above a static threshold. A response above the threshold is meant to strengthen the active synapse, and a response below the threshold should lead to a weakening of the active synapse.
One of the widely used models of synaptic plasticity is the Bienenstock–Cooper–Munro (BCM) learning rule with which Bienenstock et al. [5]—by incorporating a dynamic threshold that is a function of the average postsynaptic activity over time—captured the development of stimulus selectivity in the primary visual cortex of higher vertebrates. In corroborating the BCM theory, it has been shown that a BCM network develops orientation selectivity and ocular dominance in natural scene environments [6, 7]. Although the BCM rule was developed to model selectivity of visual cortical neurons, it has been successfully applied to other types of neurons. For instance, it has been used to explain experiencedependent plasticity in the mature somatosensory cortex [8]. Furthermore the BCM rule has been reformulated and adapted to suit various interaction environments of neural networks, including laterally interacting neurons [9, 10] and stimuli generalizing neurons [11]. The BCM rule has also been in the center of the discussion as regards the relationship between ratebased plasticity and spiketime dependent plasticity (STDP); it has been shown that the applicability of the BCM formulation is not limited to ratebased neurons but under certain conditions extends to STDPbased neurons [12–14].
Based on the BCM learning rule, a few data mining applications of neuronal selectivity have emerged. It has been shown that a BCM neural network can perform projection pursuit [7, 15, 16], i.e. it can find projections in which a data set departs from statistical normality. This is an important finding that highlights the feature detecting property of a BCM neural model. As a result, the BCM neural network has been successfully applied to some specific pattern recognition tasks. For example Bachman et al. [17] incorporated the BCM learning rule in their algorithm for classifying radar data. Intrator et al. developed an algorithm for recognizing 3D objects from 2D view by combining existing statistical feature extraction models with the BCM model [18, 19]. There has been a preliminary simulation on how the BCM learning rule has the potential to identify alpha numeric letters [20].
Mathematically speaking, the BCM learning rule is a system of differential equations involving the synaptic weights, the stimulus coming into the neuron, the activity response of the neuron to the stimulus, and the threshold for the activity. Unlike its predecessors, which use static thresholds to modulate neuronal activity, the BCM learning rule allows the threshold to be dynamic. This dynamic threshold provides stability to the learning rule, and from a biological perspective, provides homeostasis to the system. Treating the BCM learning rule as a dynamical system, this paper explores the stability properties and shows that the dynamic nature of the threshold guarantees stability only in a certain regime of homeostatic time scale. This paper also explores the stability properties as a function of the relationship between homeostasis time scale and the weight time scale. Indeed, there is no biological reason why the homeostatic time scale should be dramatically shorter than the synaptic modification time scale [21], so in this paper, we relax those restrictions. In Sect. 3, we illustrate a stochastic simulation in the simplest case of a single neuron with two weights and two different competing stimuli. We derive the averaged mean field equations and show that there are changes in the stability as the homeostatic time constant changes. In Sect. 4, we continue the study of a single neuron, but now assume that there are more inputs than weights. Here, we find rich dynamics including multiple perioddoubling cascades and chaotic dynamics. Finally, in Sect. 5, we study small linearly coupled networks and prove stability results while uncovering more rich dynamics.
2 Methods
3 Results I: One Neuron, Two Weights, Two Stimuli
For a single linear neuron that receives a stimulus pattern \(\mathbf {x}=(x_{1}, \ldots,x_{n} )\) with synaptic weights \(\mathbf{w}=(w_{1},\ldots ,w_{n})\), the neuronal response is \(v=\mathbf{w} \cdot\mathbf{x}\). The results we present in this section are specific to when \(n=2\) and when there are two patterns. In this case, the neuronal response is \(v = {w_{1}}{x_{1}} + {w_{2}}{x_{2}}\). In the next section, we explore a more general setting.
3.1 Stochastic Experiment
A good starting point in studying the dynamical properties of the BCM neuron is to explore the steady states of v for different timescale factors of θ. This is equivalent to varying the ratio \(\tau _{\theta}/ \tau_{w}\) in Eq. (2). We start with a BCM neuron that receives a stimulus input x stochastically from a set \(\{ \mathbf{x}^{(1)}, \mathbf{x}^{(2)}\}\) with equal probabilities, that is, \(Pr[\mathbf{x}(t)=\mathbf{x}^{(1)} ]=Pr[\mathbf{x}(t)=\mathbf {x}^{(2)} ]= \frac{1}{2}\). We create a simple hybrid stochastic system where the value of x switches between the pair \(\{ \mathbf{x}^{(1)}, \mathbf{x}^{(2)}\}\) at a rate λ as a two state Markov process. At steady state, the neuron is said to be selective if it yields a high response to one stimulus and a low (≈0) response to the other.
Figures 1B–D plot the neuronal response v as a function of time. In each case, the initial conditions of \(w_{1}\), \(w_{2}\) and θ lie in the interval \((0,0.3)\). The stimuli are \(\mathbf{x}^{(1)}=(\cos\alpha,\sin\alpha)\) and \(\mathbf{x}^{(2)}=(\sin\alpha,\cos\alpha)\) where \(\alpha=0.3926\). \(v_{1}= \mathbf{w} \cdot x^{(1)}\) is the response of the neuron to the stimulus \(\mathbf {x}^{(1)}\) and \(v_{2}= \mathbf{w}\cdot x^{(2)}\) is the response of the neuron to the stimulus \(\mathbf{x}^{(2)}\). In each simulation, the presentation of stimulus is a Markov process with rate \(\lambda= 5\) presentations per second. When \(\tau_{\theta}/ \tau_{w}= 0.25\), Fig. 1B shows a stable selective steady state of the neuron. At this state, \(v_{1} \approx2\) while \(v_{2}\approx0\), implying that the neuron selects \(\mathbf{x}^{(1)}\). This scenario is equivalent to one of the selective steady states demonstrated by Bienenstock et al. [5].
When the threshold, θ changes slower than the weights, w, the dynamics of the BCM neuron take on a different kind of behavior. In Fig. 1C, \(\tau_{\theta}/ \tau_{w}=1.7\). As can be seen, there is a difference between this figure and Fig. 1B. Here, the steady state of the system loses stability and a noisy oscillation appears to emerge. The neuron is still selective since there is a large enough empty intersection between these ranges of oscillation.
Setting the timescale factor of θ to be a little more than twice that of w reveals a different kind of oscillation from the one seen in Fig. 1C. In Fig. 1D where \(\tau _{\theta}/ \tau_{w}=2.5\), the oscillation has very sharp maxima and flat minima and can be described as an alternating combination of spikes and rest states. As can be seen, the neuron is not selective.
3.2 Mean Field Model
3.3 Oscillatory Properties: Simulations
As seen in the preceding section, the fixed points to the mean field BCM equation are invariant (with regards to stimuli and synaptic weights) and depend only on the probabilities with which the stimuli are presented. The stability of the selective fixed points, however, depends on the timescale parameters, the angular relationship between the stimuli, and the amplitudes of the stimuli. To get a preliminary understanding of this property of the system, consider the following simulations of Eq. (4); each with different stimulus set characteristics. We remark that because Eq. (4) depends only on the inner product of stimuli, equal rotation of both has no effect on the equations. What matters is the magnitude, angle between them, and frequency.
Simulation A: orthogonal, equal magnitudes, equal probabilities
Simulation B: nonorthogonal, equal magnitudes, equal probabilities
Let \(\rho=0.5\), \(\mathbf{x}^{(1)} = (1,0)\), \(\mathbf{x}^{(2)} = (\cos(1),\sin (1))\), \(\tau=1.5\). In this case, the two stimuli have equal magnitudes, are not perpendicular to each and are presented with equal probabilities. Figure 2(B) shows an oscillation, but now \(v_{2}\) oscillates as well since the stimuli are not orthogonal.
Simulation C: nonorthogonal, equal magnitudes, unequal probabilities
Let \(\rho=0.7\), \(\mathbf{x}^{(1)} = (1,0)\), \(\mathbf{x}^{(2)} = (\cos(1),\sin(1))\). The only difference between this case and simulation B is that the stimuli are now presented with unequal probabilities. For \(\tau=1.5\), there is a stable oscillation of both \(v_{1},v_{2}\) centered around their unstable equilibrium values.
Simulation D: orthogonal, unequal magnitude, equal probabilities
Let \(\rho=0.5\), \(\mathbf{x}^{(1)} = (1,0)\), \(\mathbf{x}^{(2)} = 1.5(\cos(1), \sin(1))\). The only difference between this case and simulation B is that stimulus 2 has a larger magnitude and \(\tau=0.8\). We remark that in this case, even when \(\tau<1\), the equilibrium point has become unstable.
These four examples demonstrate that there are oscillations of various shapes and frequencies that arise pretty generically no matter what the specifics of the mean field model are; they can occur in symmetric cases (e.g. simulation A) or with more general parameters as in BD. We also note that to get oscillatory behavior in the BCM rule, we do not even need \(\tau_{\theta}> \tau_{w}\) as seen in example D. We will see shortly that the oscillations arise from a Hopf bifurcation as the parameter, τ increases beyond a critical value. To find this value, we perform a stability analysis of the equilibria for Eq. (4).
3.4 Stability Analysis
We begin with a very general stability theorem that will allow us to compute stability for an arbitrary pair of vectors and arbitrary probabilities of presentation. Looking at Eq. (4), we see that by rescaling time, we can assume that \(\mathbf{x}^{(1)}\cdot \mathbf{x}^{(1)}=1\) without loss of generality. To simplify the calculations, we let \(\tau=\tau_{\theta}/\tau_{w}\), \(b= \mathbf{x}^{(1)}\cdot \mathbf{x}^{(2)}\), \(a=\mathbf{x}^{(2)}\cdot\mathbf{x}^{(2)}\), and \(c=\rho/(1\rho)\). Note that \(a>b^{2}\) by the Schwartz inequality and that \(c\in(0,\infty)\) with \(c=1\) being the case of equal probability.
For completeness, we first dispatch with the two nonselective equilibria, \((1,1,1)\) and \((0,0,0)\). At \((1,1,1)\), it is easy to see that the characteristic polynomial has a constant coefficient that is \(\rho(1\rho)(b^{2}a)/\tau\), which means that it is negative since \(a>b^{2}\). Thus, \((1,1,1)\) is linearly unstable.
We now use Eqs. (7) and (9) to explore the stability of the two solutions as a function of τ. We have already eliminated the possibility of losing stability through a zero eigenvalue since both \(A_{10},A_{20}\) are positive. Thus, the only way to lose stability is through a Hopf bifurcation which occurs when either of \(Q_{R}^{1,2}(\tau )\) vanishes. We can use the quadratic formula to solve for τ for each of these two cases, but one has to be careful since the coefficient of \(\tau^{2}\) vanishes when \(c=a\) or \(c=1/a\).
We summarize the results in this section with the following theorem.
Theorem 3.1

\(\mathbf{z}_{1}\) is linearly asymptotically stable if and only if$$c\bigl(ab^{2}\bigr) (1ac)\tau^{2} \bigl(1+2aca^{2}c^{2}2b^{2}c \bigr)\tau+(1+ac) > 0. $$

\(\mathbf{z}_{2}\) is linearly asymptotically stable if and only if$$c\bigl(ab^{2}\bigr) (ac)\tau^{2}+\bigl(2c \bigl(b^{2}a\bigr)+c^{2}a^{2}\bigr)\tau+ a+c > 0. $$

If \(a=1\) (that is, the stimuli have equal amplitude), then \(\mathbf{z}_{1,2}\) are linearly asymptotically stable if and only if$$c(1c) \bigl(1b^{2}\bigr)\tau^{2} +\bigl(2c \bigl(b^{2}1\bigr)+c^{2}1\bigr)\tau+ 1+c > 0. $$

In the simplest case where \(a=c=1\), then both selective equilibria are stable if and only if$$\tau< \frac{1}{1b^{2}}. $$
3.5 Bifurcation Analysis
If the amplitude of \(\mathbf{x}^{(2)}\) is different from that of \(\mathbf{x}^{(1)}\), then the theorem shows that the two selective equilibria have different stability properties. Figure 4C shows the bifurcation diagram for \(A=1.5\). When we follow the stability of \(\mathbf{z}_{1}=(2,0,2)\) (shown as the lower curve labeled 1), there is a Hopf bifurcation at \(\tau\approx1.52\) and a stable branch of periodic orbits bifurcates from it that persists up until \(\tau\approx1.94\) where it bends around (LP), becomes unstable, and terminates on the symmetric unstable equilibrium, \((v_{1},v_{2},\theta)=(1,1,1)\) at a Hopf bifurcation (\(\tau\approx0.79\)) for this equilibrium, labeled Hs. Figure 4D shows the small amplitude periodic orbit at \(\tau=1.7\) projected in the \((v_{1},v_{2})\) plane where it is centered around \((v_{1},v_{2})=(2,0)\). The upper curve in panel C (labeled 2) shows the stability of \(\mathbf{z}_{2}=(0,2,2)\) as τ varies. Here, there is a Hopf bifurcation at \(\tau\approx0,5\) and a stable branch of periodic orbits bifurcates from the equilibrium. The branch terminates at a homoclinic orbit at \(\tau\approx1.35\). Figure 4D shows an orbit for \(\tau=0.7\) that surrounds \((v_{1},v_{2})=(2,0)\).
In sum, in this section we have analyzed a very simple BCM model where there are two stimuli, two weights, and one neuron. We have shown that if the timescale factor (\(\tau_{\theta}\)) of the homeostatic threshold, θ is too slow relative to the timescale factor of the weights, then, the selective equilibria lose stability via a Hopf bifurcation and limit cycles emerge. For very large ratios, \(\tau=\tau_{\theta}/\tau _{w}\), solutions become unbounded and intermediate values of τ, the origin becomes an attractor even though it is unstable. In the next section, we consider the case when there are more stimuli than there are weights and, in the subsequent section, we consider small coupled networks.
4 Results II: One Neuron, n Weights, m Stimuli
4.1 Example: \(n=2,m=3\)
Figure 6 shows some probable chaos for \(\tau=1.8\) and \(C\in[0,0.25]\). Panel A shows a trajectory projected in the \(v_{1}\theta \) plane for \(C=0.18\). Panel B shows the evolution of the attracting dynamics as C varies. We take a Poincaré section at \(v_{2}=2\) and plot the successive values of θ after removing transients and letting C vary between 0 and 0.25. As C increases, there is a periodic orbit that undergoes multiple perioddoubling bifurcations before becoming chaotic. There are several regions showing period three orbits (\(C\approx0.1\), \(C\approx0.175\), \(C\approx0.21\)) as well as many regions with complex behavior. The chaos and periodic dynamics terminates near \(C=0.237\), which is the value of C at which the lower stable branch of equilibria in the isola begins. Chaos and similar complex dynamics occurs for other values of τ.
In this section, we have shown that the degeneracy that occurs when there are more stimulus patterns than weights can be resolved by finding some simple constants of motion. The resulting reduced system will always be threedimensional. In the simplest case of three patterns and two weights, we have found rich dynamics when \(\tau=\tau _{\theta}/\tau_{w}\) is larger than 1.
5 Results III: Small Coupled Network
To implement a network of coupled BCM neurons receiving stimulus patterns from a common set, it is important to incorporate a mechanism for competitive selectivity within the network. A mechanism of this sort, found in visual processes [23] (and also in tactile [24], auditory [25], and olfactory processing [26]) is called lateral inhibition, during which an excited neuron reduces the activity of its neighbors by disabling the spreading of action potentials to neighboring neurons in the lateral direction. This creates a contrast in stimulation that allows increased sensory perception.
5.1 Mean Field Model
Solving this system of equations gives the set of fixed points \((v_{a1}, v_{a2}, \theta_{a}, v_{b1}, v_{b2}, \theta_{b}) = \{(0, 0, 0, 0, 0, 0), (\frac{1}{\rho}, 0, \frac{1}{\rho} , \frac{1}{\rho}, 0, \frac {1}{\rho} ), ( 0, \frac{1}{1\rho}, \frac{1}{1\rho}, 0, \frac {1}{1\rho}, \frac{1}{1\rho} ), (\frac{1}{\rho}, 0, \frac{1}{\rho} , 0, \frac{1}{1\rho}, \frac {1}{1\rho} ), (0, \frac{1}{1\rho}, \frac{1}{1\rho} ,\frac{1}{\rho}, 0, \frac{1}{\rho} ), (1, 1, 1, 1, 1,1), (1,1,1,\frac{1}{\rho},0,\frac{1}{\rho}), \ldots\}\). The … in these fixed points correspond to the symmetric variants of the last equilibrium, for example swapping the \((1,1,1)\) and the \((\frac{1}{\rho},0,\frac{1}{\rho})\) or swapping the latter triplet for \((0,\frac{1}{1\rho},\frac{1}{1\rho})\).
Castellani et al. [9] and Intrator and Cooper [7] give a detailed analysis on the stability of most of these fixed points in the limit of \(\tau_{\theta}\to0\). They showed that \((0,0,0, 0,0,0)\) and \((1,1,1, 1,1,1)\) are unstable and the fully selective fixed points are stable. This leaves the fixed points of the form \((1,1,1,\frac {1}{\rho},0,\frac{1}{\rho})\). We address these below for our particular choice of stimuli.
We will explore the dynamics of Eq. (22) as \(\tau =\tau _{\theta}/\tau_{w}\) changes in a very simple scenario in which, \(\rho=0.5\), and \(\mathbf{x}^{(1)}=(1,0)\) and \(\mathbf{x}^{(2)}=(\cos\alpha,\sin \alpha)\). In this case, there are only two equilibria that need to be studied: the symmetric case \((\theta_{a},v_{a1},v_{a2},\theta _{b},v_{b1},v_{b2})= (2,2,0,2,2,0)\) and the antisymmetric case, \((2,2,0,2,0,2)\). The other selective equilibria are symmetric to these two. In the symmetric case, neurons a and b are both selective to stimulus 1 and in the antisymmetric case, neuron a selects stimulus 1 and neuron b selects stimulus 2. Fixing α, the angle between the stimuli leaves two free parameters, τ and γ, the inhibitory coupling.
5.2 Stability of the Selective Equilibria
We summarize the stability results in the following theorem.
Theorem 5.1
 1.The symmetric equilibrium, \((v_{a1},v_{a2},\theta _{a},v_{b1},v_{b2},\theta_{b})=(2,0,2,2,0,2)\) is linearly asymptotically stable if and only ifFurthermore the unstable direction is antisymmentric.$$\tau< \tau_{H}^{s} = \frac{1\gamma}{1\cos^{2}(\alpha)}. $$
 2.The antisymmetric equilibrium \((v_{a1},v_{a2},\theta _{a},v_{b1},v_{b2},\theta_{b})=(2,0,2,0,2,2)\) is linearly asymptotically stable if and only ifFurthermore the unstable direction is symmetric.$$\tau< \tau_{H}^{a} = \frac{1\gamma\cos(\alpha)}{1\cos^{2}(\alpha)}. $$
We remark that, for acute angles where \(\cos(\alpha)>0\), the symmetric equilibrium loses stability at lower values of τ than does the antisymmetric equilibrium and for obtuse angles (\(\cos\alpha<0\)) it is vice versa.
5.3 Numerical Results
In this section, we study the numerical behavior of Eq. (17) for \(\rho=0.5\), \(\mathbf{x}^{1,2}\) unit vectors with angle \(\alpha=0.7709\) as τ and γ vary. We will generally set \(\gamma=0.25\). The choice for α is somewhat arbitrary but was found to yield rich dynamics.
6 Discussion
We have explored the BCM rule as a dynamical system. Although the literature does not suggest a homeostatic timescale range that ensures stability of a biological system, we have shown that the selective fixed points of the BCM rule are generally stable when the homeostatic time scale is faster than synaptic modification time scale, and that some complex dynamics emerges as the homeostatic time scale varies. The nature of this complex dynamics also depends on the angular and amplitudinal relationships between stimuli in the stimulus set. In our analysis, the neuron is presented with stimuli that switch rapidly, so it was possible to reduce the learning rule to a simple averaged set of differential equations. We studied the dynamics and bifurcation structures of these averaged equations when the homeostatic time scale is close to the synaptic modification time scale, and found that instabilities arise, leading to oscillations and in some cases chaos and other complex dynamics. Similar results would hold if the quadratic term \(v^{2}\) in the second line of Eq. (2) were replaced with \(v^{p}, p>2\), since the original formulation by Bienenstock et al. [5] suggests that the fixed point structures are preserved for any positive value of p. Since the onset of the bifurcations (such as the Hopf bifurcation) depends mainly on the symmetry of these fixed points, we expect that the main results will be the same and only the particular values of parameters would change. While this paper has focused on how small changes in the time scale of a homeostatic threshold can lead to complex dynamics, there are many other kinds of homeostases [28] which present many time scales and similar opportunities for analysis.
The model neuron we used in this paper has been assumed to have linear response properties, which may be seen as oversimplified, and hence a potential problem in translating our conclusions to actual biological systems. It is well known that plasticity goes beyond synapses, and it is sometimes even a neuronwide phenomenon [29], and that there is no unique route to regulating the sliding threshold of the BCM rule [10, 30]. Thus in addition to synaptic activities, intrinsic neuronal properties may also play a role in the evolution of responses and linearity may not be able to capture this scenario. The introduction of a nonlinear transfer function to the BCM learning rule has been addressed by Intrator and Cooper [7]. In their formulation, the learning rule is derived as a gradient descent rule on an objective function that is cubic in the nonlinear response. Our decision to use linear units is motivated by the accessibility to formal analysis. Biologically, linearity can be justified if we assume that the underlying biochemical mechanisms are governed by membrane voltage rather than firing rate; see, for example Clopath and Gerstner [31].
The theoretical contributions of this paper are based on an analysis that we did using a mean field model of the BCM learning rule. Similar mean field models have been made, but in terms of synaptic weights; see, for example Yger and Gilson [32]. With this approach to the mean field, it is difficult to arrive at the fact that the fixed points—not their stability—of the learning rule depend only on the probabilities with which each stimulus is presented. In this paper, we have given a derivation of the mean field model of the BCM learning rule as a rate of change of the activity response v, with time. The derived model considers the amplitudes of the stimuli presented, the pairwise angular relationships between the stimuli, and the probabilities with which the stimuli are presented. The appeal of this derivation is that it easily highlights the fact that the fixed points depend on these probabilities. Additionally, the derivation is important because the dynamics of the BCM learning rule is driven by the activity response (not the synaptic weights), and many analyses in the literature rely on this fact; see, for example Castellani et al. [9]. Our analyses considered three cases: one neuron with two weights and two stimuli, one neuron with two weights and three stimuli, and lastly a weakly interacting small network of neurons.
In our analysis of a small network (see Sect. 5) we have made the simplifying assumption that the lateral inhibitory weight is constant in time. The incorporation of an inhibitory plasticity rule (as in Moldakarimov et al. [35]) would necessitate a third timescale parameter, and possibly a fourth if the inhibitory rule were to include a dynamic modification threshold. This is beyond the scope of the paper and reserved for future work. Another related possible future direction is to perform an analysis of a large network of BCM neurons, by observing what happens to the network dynamics at different timescale parametric regimes. A good starting point is to explore the dynamics for a fully connected network with equal inhibition, that is, each neuron is coupled with every other neuron in the network and inhibits each of them equally. The next step would be to let the amount of inhibition vary according to how far away the inhibiting neuron is. It may also be interesting to examine how the architecture of the network is affected. We know, for instance, that spiketime dependent plasticity (STDP) has the ability to yield a feedforward network out of a fully connected network. The analysis that Kozloski and Cecchi [36] used to demonstrate this finding centers around the synaptic weights. Thus it will be useful to pay closer attention to the synaptic weights in future work. Moreover, the oscillatory and chaotic properties we observed in the small coupled network will also be observed had our mean field been derived in terms of the weight and the analyses been done with the synaptic weights.
The debate about synaptic homeostatic time scales in neurobiology remains vibrant. A review of the literature seems to reveal a varied, and somewhat paradoxical set of findings among experimentalists and theoreticians. While homeostasis of synapses found in experiments is slow [12, 37], homeostasis of synapses in most theoretical models needs to be rapid and sometimes even instantaneous to achieve stability [33, 38, 39]. There are, however, ongoing efforts to shed more lights on the debate. It has been suggested that both fast and slow homeostatic mechanisms exist. Zenke and Gerstner [39] suggest that learning and memory use an interplay of both forms of homeostasis; while fast homeostatic control mechanisms help maintain the stability of synaptic plasticity, slower ones are important for finetuning neural circuits. In addition to the present work contributing to the debate by demonstrating the relevance of fast homeostasis to synaptic stability, it also furthers the discussion as regards the link between STDP and the BCM rule: Zenke et al. [33] found that homeostasis needs to have a faster rate of change for spiketiming dependent plasticity to achieve stability. Furthermore it is well known that, under certain conditions, the BCM learning rule follows directly from STDP [13, 14].
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Hebb D. The organization of behavior. New York: Wiley; 1949. Google Scholar
 Hertz J, Krogh A, Palmer R. Introduction to the theory of neural computation. Reading: AddisonWesley; 1991. Google Scholar
 Nass MN, Cooper L. A theory for the development of feature detecting cells in the visual cortex. Biol Cybern. 1975;19:1–18. View ArticleGoogle Scholar
 Cooper LN, Liberman F, Oja E. A theory for the acquisition and loss of neuron specificity in the visual cortex. Biol Cybern. 1979;33:9–28. MathSciNetView ArticleMATHGoogle Scholar
 Bienenstock EL, Cooper L, Munro P. Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex. J Neurosci. 1982;2:32–48. Google Scholar
 Shouval H, Intrator N, Cooper L. BCM network develops orientation selectivity and ocular dominance in natural scene environment. Vis Res. 1997;37(23):3339–42. View ArticleGoogle Scholar
 Intrator N, Cooper L. Objective function formulation of the BCM theory for visual cortical plasticity: statistical connections, stability conditions. Neural Netw. 1992;5:3–17. View ArticleGoogle Scholar
 Bliem B, MuellerDahlbaus JFM, Dinse HR, Ziemann U. Homeostatic metaplasticity in human somatosensory cortex. J Cogn Neurosci. 2008;20:1517–28. View ArticleGoogle Scholar
 Castellani GC, Intrator N, Shouval H, Cooper L. Solutions of the BCM learning rule in a network of lateral interacting nonlinear neurons. Netw Comput Neural Syst. 1999;10:111–21. View ArticleMATHGoogle Scholar
 Cooper LN, Bear MF. The BCM theory of synapse modification at 30: interaction of theory with experiment. Nature. 2012;13:798–810. Google Scholar
 Munro PW. A model for generalization and specification by a single neuron. Biol Cybern. 1984;51:169–79. View ArticleMATHGoogle Scholar
 Yeung LC, Shouval HC, Blais BS, Cooper LN. Synaptic homeostasis and input selectivity follow from a calciumdependent plasticity model. Proc Natl Acad Sci USA. 2004;101(41):14943–8. View ArticleGoogle Scholar
 Izhikevich EM, Desia N. Relating STDP to BCM. Neural Comput. 2003;15:1511–23. View ArticleMATHGoogle Scholar
 Gjorgjieva J, Clopath C, Audet J, Pfister JP. A triplet spiketiming dependent plasticity model generalizes the Bienenstock–Cooper–Munro rule to higherorder spatiotemporal correlations. Proc Natl Acad Sci USA. 2011;108(48):19383–8. View ArticleGoogle Scholar
 Dotan Y, Intrator N. Multimodality exploration by an unsupervised projection pursuit neural network. IEEE Trans Neural Netw. 1998;9:464–72. View ArticleGoogle Scholar
 Intrator N, Gold JI. Threedimension object recognition of gray level images: the usefulness of distinguishing features. Neural Comput. 1993;5:61–74. View ArticleGoogle Scholar
 Bachman CM, Musman S, Luong D, Schultz A. Unsupervised BCM projection pursuit algorithms for classification of simulated radar presentations. Neural Netw. 1994;7:709–28. View ArticleGoogle Scholar
 Intrator N. Feature extraction using unsupervised neural network. Neural Comput. 1992;4:98–107. View ArticleGoogle Scholar
 Intrator N, Gold JI, Bülthoff HH, Edelman S. Threedimensional object recognition using an unsupervised neural network: understanding the distinguishing features. In: Proceedings of the 8th Israeli Conference on AICV. 1991. Google Scholar
 Poljovka A, Benuskova L. Pattern classification with the BCM neural network. In: Stopjakova V, editor. Proc. 2nd Electronic Circuits and Systems Conference—ECS’99, Bratislava. 1999. p. 207–10. Google Scholar
 Turrigiano GG, Nelson SB. Homeostatic plasticity in the developing nervous system. Nat Rev Neurosci. 2004;5(2):97–107. View ArticleGoogle Scholar
 Dayan P, Abbott L. Theoretical neuroscience: computational and mathematical modeling of neural systems. Cambridge: MIT Press; 2001. MATHGoogle Scholar
 Field G, Chichilnisky E. Information processing in the primate retina: circuitry and coding. Annu Rev Neurosci. 2007;30:1–30. View ArticleGoogle Scholar
 Johansson R, Vallbo AB. Tactile sensory coding in the glabrous skin of the human hand. In: TINS. 1983. Google Scholar
 Pantev C, Okamoto H, Ross B, Stoll W, CiurliaGuy E, Kakigi R, Kubo T. Lateral inhibition and habituation of the human auditory cortex. Eur J Neurosci. 2004;19(8):2337–44. View ArticleGoogle Scholar
 Yantis S. Sensation and perception. New York, NY: Worth Publishers; 2013. Google Scholar
 Ermentrout GB, Terman DH. Mathematical foundations of neuroscience. Berlin: Springer; 2010. View ArticleMATHGoogle Scholar
 Gjorgjieva J, Drion G, Marder E. Computational implications of biophysical diversity and multiple timescales in neurons and synapses for circuit performance. Curr Opin Neurobiol. 2016;37:44–52. View ArticleGoogle Scholar
 Zhang W, Linden DJ. The other side of the engram: experiencedriven changes in neuronal intrinsic excitability. Nat Rev Neurosci. 2003;4:885–900. View ArticleGoogle Scholar
 Anirudhan A, Narayanan R. Analogous synaptic plasticity profiles emerge from disparate channel combinations. J Neurosci. 2015;35(11):4691–705. View ArticleGoogle Scholar
 Clopath C, Gerstner W. Voltage and spike timing interact in STDP—a unified model. Front Synaptic Neurosci. 2010;2:25. Google Scholar
 Yger P, Gilson M. Models of metaplasticity: a review of concepts. Front Comput Neurosci. 2015;9:138. View ArticleGoogle Scholar
 Zenke F, Hennequin G, Gerstner W. Synaptic plasticity in neural networks needs homeostasis with a fast rate detector. PLoS Comput Biol. 2013;9(11):e1003330. View ArticleGoogle Scholar
 Toyoizumi T, Kaneko M, Stryker MP, Miller KD. Modeling the dynamic interaction of Hebbian and homeostatic plasticity. Neuron. 2014;84(2):497–510. View ArticleGoogle Scholar
 Moldakarimov SB, McClelland JL, Ermentrout GB. A homeostatic rule for inhibitory synapses promotes temporal sharpening and cortical reorganization. Proc Natl Acad Sci USA. 2006;103(44):16526–31. View ArticleGoogle Scholar
 Kozloski J, Cecchi G. A theory of loop formation and elimination by spike timingdependent plasticity. Front Neural Circuits. 2010;4:7. Google Scholar
 Turrigiano GG, Nelson SB. Hebb and homeostasis in neuronal plasticity. Curr Opin Neurobiol. 2000;10(3):358–64. View ArticleGoogle Scholar
 Miller KD, MacKay DJC. The role of constraints in Hebbian learning. Neural Comput. 1994;6:100–26. View ArticleGoogle Scholar
 Zenke F, Gerstner W. Hebbian plasticity requires compensatory processes on multiple timescales. Philos Trans R Soc Lond B. 2016;372:20160259. View ArticleGoogle Scholar