Basic Categories of Synaptic Learning Rule
- Associative vs non-associative
- Non associative, involves one side only.
- Associative, involves activity of both pre and post. Associative rules are more often used in computation.
- Local vs Global (homo, hetero synapse)
- Hetero synapse is to modify weight $w_{B\to A}$ by synapse $w_{C\to A}$.
- Direction: Potentiation, Depression
- Time scale: Short term, Long term
Hebbian Learning
$$ \tau_w dw/dt = u v $$Updated version is this covariance rule or extended Hebbian Learning
$$ \tau_w \frac{dw}{dt} = u (v-- synapse will depress or excite based on activity of post syn firing. Weight will increase if post syn firing higher than average.
These learning rules are covariance based.
A more extended version is the BCM rule (Bienenstock, Cooper, and Munro learning rule). Tune up or down the synapse weight based on a nonlinear function of post-syn activity.
$$ \tau_w \frac{dw}{dt} = u\cdot \phi(v,\Theta) $$- $\phi(v)<0$ if $v<\Theta$ , $\phi(v)>0$ if $v>\Theta$
- $\Theta = \bar v(t)$ provide an adaptive threshold similar to covariance rule. $\bar v(t)$ is a time average of recent post-syn activity. Moving average.
- This define a dynamic system of $w$ and $\Theta$
- At first, weak synapses make post syn v has low firing rate. $\Theta$ is low. Thus all weights tend to grow.
- After weight grows, $\Theta$ gets higher, then only above average input patterns are facilitated, not all.
- Finally the environment or distribution of the inputs $u$ determines what it learns.
There are theoretical results for this synaptics dynamics
Mathematical results, obtained only for certain discrete distributions, are of two types: (2) equilibrium points are locally stable if and only if they are of the highest available selectivity with respect to the given distribution of d and (2) given any initial value of m in the state space, the probability that m(t) converges to one of the maximum selectivity fixed points as t goes to infinity is 1.
Weight explosion
- Using only Hebbian or extended Hebbian rule, all weights will explode.
- Need criterion for weight decay.
Oja’s Rule
It adds an intrinsic weight decay into the dynamics, depending on the input strength: Stronger input, faster decay. ($v^2$ is element-wise square amplitude of activity.)
$$ \tau_w\frac{dw}{dt}=vu-\alpha v^2 w $$L2 Norm of weight value will stay around a fixed point $\|w\|^2$
This can predict or model the homeostasis or competition of synaptic weights: when one synapse strengthen a lot, the other synapses weaken to compensate.
BCM Rule
Remarks
- Note, Hebbian learning requires there are some synapse there in the first place. No weights or 0 weights, will result in no activity => no learning could happen.
- This is the same for deep learning. If initial weight is 0, back propagation cannot work.
- Biologically, there need to be some synapse to be learned.
Content of Hebbian Learning
Hebbian learning is learning the first PC.
If we average over different inputs, the Hebbian learning equation becomes
$$ \tau_w \frac{dw}{dt} = Cw\\ C=<(u-)u> $$