The Role of Constraints in Hebbian Learning Miller and MacKay 1994, Neural Comput: 101-126

1. The Role of Constraints in Hebbian Learning Miller and MacKay 1994, Neural Comput: 101-126

2. Outline • Constraints on Hebbian Plasticity: • Importance • Types of Constraints • Dynamic Effects of Constraints • Biological evidence for Constraints • Function of Constraints in Heterogeneous Networks

3. Hebb’s Rule τw dw/dt = vu = C·w (Correlation based learning rule) where C = uu or the input correlation matrix • Problem with Hebb’s Rule: • weights grow without bounds => instability • loss of selectivity to different patterns of input

4. Solution: Introduce Competition (constraint that limits total synaptic strength over cell) • Multiplicative: synapse decays at rate proportional to its current strength • Subtractive: synapse decays at a fixed rate where n = (1,1,….1)T in the synaptic basis

5. Types of Constraints the two methods can enforce: • Type 1 – Conserve total synaptic strength • or Hence, Hyperplane Constraint Surface

6. Type 2 – Conserve sum-squared synaptic strength Hence, M2: γ(w) = w · Cw/ w · w Hypersphere Constraint Surface

7. Dynamic Effects of Multiplicative and Subtractive Constraints +ve correlation in Hebb’s rule (e0 is close in direction to constraint vector n) Cw α n Cw α w Correlations oscillate in sign (e0 || to constraint surface e0·n = 0 e0 treated as zero sum vector) e0 always  constrained surface

8. Theorem 1: Under a multiplicatively enforced constraint, if principal eigen vector of C is an interior fixed point it is stable. Interior fixed points that are nonprincipal eigenvectors are unstable. (saturation unnecessary) Theorem 2: Under an S1 constraint, if C has atleast two eigenvectors with positive eigenvalues, then any interior fixed point is unstable. Theorem 3: If i and j are indices in the synaptic basis, and Cii > |Cij| then under S1 constraint, either all synapses (or all but one) are saturated in a stable final condition. S1 constraints lead to a zero sum vector that grows to complete saturation.

9. Outcomes of development with and without Constraints RF has graded strengths RFs sharpened; Ocular dominance develops (final number of non-zero synapses α wtot/wmax)

10. S1 constraints more appropriate to model Ocular Dominance than M1 constraints Let w1 and w2 be the synaptic weight vector from each input projection ws = w1 + w2 wd = w1 – w2 As inputs are symmetric, eigenvectors can be divided into sum eigenvectors: ws = eSa, wd = 0 (eigenval λSa) and wd = eDa, ws = 0 (eigenval λDa) Patterns of wd have zero total synaptic strength => wd grows freely under S1; wd suppressed under M1 (unless eDa is the principal eigenvector i.e. λDa > λSa – only possible with –ve correlations between inputs from two eyes!)

11. Constraints applied to a full layer of output cells – Heuristic Approach

12. Suggested Biological Implementation of Subtractive Constraints: • Limited capacity of metabolic supply to synapses (constant decay imposed on each synapse) • decay rate dependent on average degree of activation of cell However, S1 punishes weak synapses more than M1

13. Evidence for another type of Constraint : Multiplicative Homeostatic Scaling in Cultured Networks Turrigiano and Nelson, Nature Rev Neuro. 5: 97-107 (2004)

14. Physiology of Homeostatic Scaling ___ multiplicative scaling ….. random additive scaling ----- additive scaling Turrigiano et al, Nature 391: 892-896 (1998)

15. Mechanisms of Homeostatic Scaling? Network level Single cell level

16. Renart et al, Neuron 38: 473-485 (2004) Heterogeneous Cell Properties Homeostatic Scaling applied to Spatial Working Memory Homogeneous Cell Properties

17. Homeostatic Scaling allows robust Spatial Memory Encoding in Heterogeneous Networks Isyn = gssgsyn(V-Vsyn)

18. Summary • Constraints are required to clamp uncontrolled growth of Hebbian plasticity and to maintain input selectivity • Types of Constraints – S1, M1, M2 • Biological evidence for multiplicative scaling constraint • Multiplicative homeostatic scaling allows robust encoding in heterogeneous networks