Time-Skew Hebb Rule in a Nonisopotential Neuron Barak A. Pearlmutter 
To appear (1995) in Neural Computation, 7(4) 706–712 Abstract
In an isopotential neuron with rapid response, it has been shown that the receptive fields formed by Hebbian synaptic modulation depend on the principal eigenspace of ✁ (0), the input autocorrelation matrix, where Qij(τ) = ξi(t) ξj(t τ) and ξi(t) is the input to synapse i at time t (Oja 1982). We relax the assumption of isopotentiality, introduce a time-skewed Hebb rule, andfind that the dynamics of synaptic evolution are determined by the principal eigenspace of ✁ .
(Qij ψi)(τ) Kij(τ) dτ, where Kij(τ) is the neuron’s voltage response to a unit current injection at synapse j as measured τ seconds later at synapse i, and ψi(τ) is the time course of the opportunity for modulation of synapse i following the arrival of 1 Introduction
Hebbian synaptic modification involves the enhancement of synaptic efficacy in response to simul-taneous pre- and post-synaptic activity. This form of learning has taken on particular importancewith its discovery in various parts of the brain, a prominent example being long-term potentiation(LTP) in the hippocampus (Bliss and Collingridge 1993).
The classic Oja (1982) analysis of the temporal evolution of Hebbian synapses assumes depen- dence on instantaneous conjunction of pre- and postsynaptic activity and a linear neuron with anisopotential membrane which responds quickly compared to the time course of its inputs. How-ever, it is believed that the trans-membrane potential in cortical neurons can vary widely across thedendritic tree on a time scale important to LTP (Koch et al. 1982; Rall 1977; Zador et al. 1995).
It also appears that LTP occurs when postsynaptic voltage is elevated during a finite time windowfollowing presynaptic activation (Bliss and Collingridge 1993). This paper analyzes the effects ofHebbian learning under these conditions, and grew out of an attempt to understand the clusteredspatial structure of facilitated Hebbian synapses seen in the model nonisopotential neurons of Brownet al. (1991) and Mel (1992). Even though linearity in Hebbian systems can be less restrictive thanone might suppose (Miller 1990), it is important to emphasize that the analysis here concerns aneuron that is nonisopotential but nonetheless linear. Nonlinear channels and saturation1 are notconsidered, although they likely play an important computational role (Mel 1993).
Siemens Corporate Research, 755 College Road East, Princeton, NJ 08540, bap@learning.scr.siemens.com 1We model synaptic events as current injections,even though it would be more faithful to the biology to use conductance changes. Unfortunately, using conductancechances instead of current injection introduces the solution of a general causallinear integral equation into the calculation of ✁ .
2 Results
In Oja (1982) and its extensions, the post-synaptic potential V(t) is shared across all synapses,and the input signal ξ(t) plays two roles: that of presynaptic activity in V(t) = ☛ j wj ξj(t), where wj is the efficacy of synapse j, and that of the window of opportunity for synaptic facilitation in dwi dt = ηξi(t) V(t) decay, where the learning rate η > 0 is a constant of proportionality. Taking the expected value, this leads to an expected dynamics of d decay , where
Qij(τ) = ξi(t) ξj(t τ) . These expected dynamics approximate the true dynamics well when η is small. The synaptic evolution is therefore governed2 by the principal eigenvectors of ✁ (0).
Here, we allow the neuron to be nonisopotential with ξj(t τ) Kij(τ) dτ = where Vi(t) is the post-synaptic potential at synapse i, Kij(τ) gives the voltage response of the neuron to a unit current injection at synapse j as measured τ seconds later at synapse i, andfunctional convolution is defined by (f g)(t) = τ) g(τ) dτ. The transfer impedence matrix (Koch et al. 1982) is easy to calculate and reason about in the temporal frequency domain (Zador We also generalize the Hebb rule by distinguishing between ξi(t), the presynapic input3 to synapse i at time t, and ψi(τ), the window of opportunity for synaptic modulation following presynaptic activity at synapse i. The synaptic modulation equation thus becomes i(τ) dτ Vi(t) = η (ξi ψi)(t) Vi(t) decay Kij(τ) (Qij ψi)(τ) dτ Kij(τ) (Qij ψi)(τ) dτ then, in matrix notation, we obtain the familiar 2When using a simple Hebb rule in a linear neuron, it is necessary to invoke an additional mechanism to prevent runaway synaptic facilitation. The particular decay term used has a strong effect, an issue explored in Oja (1982), Miller
and MacKay (1994), and Goodhill and Barrow (1994). In essence, with a multipicative decay term, w evolves to the
principal eigenvector of Q(0) regardless of its initial value, whereas subtractive decay terms cause w to evolve towards
principal eigenvectors of Q(0) but also tend towards the corners of the space.
3To speak of it concretely, ξi(t) typically consists of a series of delta pulses of constant magnitude representing the arrival of action potentials at the presynaptic terminal. Presynaptic changes do not affect ξi(t)—rather, wi encapsulatesall the synaptic efficacy, regardless of whether changes to that efficacy are implemented by pre- or post-synaptic physical where ✁ has replaced the usual instantaneous autocorrelation matrix ✁ (0). Thus, the evolution is determined by the principal eigenspace of ✁ in exactly the same way that, in the classic case, its evolution is determined by the the principal eigenspace of ✁ (0). Note that unlike ✁ (0), ingeneral ✁ is not symmetric. However, ✁ takes on simplified forms under important special cases, such as when ψi is a simple delay, when the neuron is isopotential, or when the neuron’s response time is fast compared to the time course of its input.
3 Application
Given known electrotonic structure, known correlational structure of the input, and known conditionsunder which synapses are modulated, one can calculate ✁ , and thus it’s principal components, and thereby predict the receptive field patterns that will be stabilized. When the input and electrotonicstructure are particularly simple, it becomes possible to calculate the principal components of ✁✆analytically. (Broad-band synaptic input with 1 f d correlational structure in both time and space impinging on either an infinite cable or an infinite sheet makes ✁ ’s principal eigenvectors particularly tractable.) Thus one can predict the formation of clusters, and their length scale, in a manner similarthat used by Miller, Keller, and Stryker (1989) to account for receptive fields in the visual system,and precisely analogous to that used by Chernjavski and Moody (1990) to predict the length scale ofcortical columns. This is similar in spirit to Mainen et al.’s (1991) account of synapse segregationon nonisopotential neurons, but operates analytically, without laborious simulation. Unfortunately,by tuning within the region of physiological parameter space consistent with experimental data, itis possible to obtain almost any length scale.
We will therefore apply the technique to a simpler situation, where it predicts a robust qualitative effect. In figure 1 a simple model neuron is constructed. The two synapses are given uncorrelatedspike trains for input. We let ψ1 = ψ2 be a square wave of duration comparable to the inter- spike interval. The elements Qij ψ are therefore all equal. This means that, up to a constant, (τ) dτ. This matrix is trivially computed, since Qij is simply the steady-state voltage at synapse i in response to a constant unit current injected at synapse j. The two elements of theprincipal eigenvector of this matrix predict the equilibrium values of the two synaptic weights. Inplotting these two synaptic weights as a function of the diameter of the soma (figure 2) we noticean interesting effect: when the soma is absent (D = 0) the symmetry of the situation is unbroken,so the two synaptic weights converge to the same value. As D is raised, the soma begins to actas a current sink. This sink is more effective at the more proximal synapse, where the Hebb rulebecomes less effective due to the lowered postsynaptic potential. This results in a strengthening ofthe distal synapse at the expense of the proximal one.
The effect levels off, because even were the soma a perfect current sink, the dendritic resistance R23 holds potential in the proximal dendritic compartment long enough for some synaptic modulation 4 Summary and Conclusion
We have shown that passive nonisopotential time-skewed Hebbian learning is mathematically anal-ogous to the isopotential instantaneous case, in that both take the form of an iterated linear operator,plus a decay term, d decay . The destiny of is thus determined by the principal
eigenspace of ✁ and the nature of the decay term, and techniques that have been applied to the classic case can now be applied to the somewhat more realistic neurons and Hebb rule discussedhere.
I would like to thank Ted Carnevale, Uzi Levin, Zach Mainen, Bartlett Mel, John Moody, andespecially Tony Zador and Ken Miller for helpful comments and suggestions.
submission of this paper, I found that results similar to those in Pearlmutter and Brown (1992), apreliminary abstract of this paper, had been independently discovered by John E. Moody and AlexChernjavski in 1989, but were not published. Portions of this work were supported by grant ONR067-32123-332 to Thomas H. Brown and by grants NSF ECS-9114333 and ONR N00014-92-J-4062to John Moody.
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Figure 1: A three-compartment model neuron with two synaptic inputs. The equivalent circuitparameters are therefore C1 = C2 = πdLCm, C3 = πD2Cm, R1 = R2 = Rm πdL, R3 = Rm πD2, R12 = R23 = 4RaL πD2. Membrane parameters used were Cm = 1µF cm2, Rm = 50kΩcm2, Ra = 200Ωcm, d = 2µm, L = 100µm. D was allowed to vary.
Figure 2: The distal (upper) and proximal (lower) final synaptic weights as a function of D, thesomatic diameter in cm. These predicted final weights exactly match the final weights evolved insimulations of this process, which were conducted at D = 0, 0.002, 0.004, 0.01cm.

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