Fast and slow homeostatic plasticity
The issue of timescale is of great importance to homeostatic plasticity. While experiments reported mechanisms of slow timescales (a timescale of hours or days), theoretical work regarding Hebbian learning requires fast time scales to maintain stability (typically around the same time scale as the Hebbian rule). This paper from Dr. Zenke and Gerstner takes us through a journey of why rapid forms of homeostasis is necessary theory-wise, then argues step-by-step what the potential functions of homeostasis mechanisms with different timescales could be. There is also a nice introduction to classification of synaptic plasticity as well as the notion of time scales, therefore I would highly recommend those unfamiliar with these topics to check out their original paper.
To begin with, let’s start with a simple form of Hebbian learning, (wij ) ̇=ηxjyi, where wij is the synaptic weight from the jth pre-synaptic neuron to the ith post-synaptic neuron, and xj and yi are the firing rates of the pre- and post-synaptic neurons respectively. Note that this differential equation is inherently unstable, as it is a positive feedback loop with no mechanisms that would either counteract or bound it. Hence this is not biologically plausible (nor useful) at all. It is therefore necessary to introduce other terms that can stabilize it. The two main forms that can do that is 1) introducing a sliding threshold and 2) adding an independent term for rate control. The “threshold” here means that if the firing rate of the neuron is lower than some target value θ, then LTD is induced; if it is higher, then LTP is induced. A “sliding” threshold is therefore one that changes its values according to local variables, and is usually assumed to depend on yi. An example of Hebbian learning plus a sliding threshold is (wij ) ̇=ηxjy(yi-θ(yi)), where (yi) is the low-pass filtered yi, representing a time-averaged like term. This helps stability in the sense that if the firing rate is too high, θ will react accordingly, therefore forcing the weight change to go from LTP to LTD. The reverse is true as well.
The rate control term, on the other hand, may take the form of η ̃(κ-(yi ))wij. η ̃ here is it’s learning rate, and κ is the target value for firing rate. Deviations from it will induce a change in the opposite direction, thereby achieving stability as well. The interesting thing about both mechanisms is that they need to be on a time scale that is fast enough. Therefore, while it has not been shown experimentally that rapid homeostasis mechanisms exist at all, it is quite necessary for theoretical work if Hebbian learning is involved.
The next issue even after we accept the premise that one of these forms exist, is that the equation is monostable. Having a monostable system means that it is hard to encode information, since all weights eventually converge to the same attractor. The solution to that is to implement multiple set points, or even a target range. There are many forms of homeostasis that can do this, and the authors of this paper argue that Hebbian plus non-Hebbian plasticity mechanisms, when in the right parameter range, naturally gives rise to bistability.
The final issue that they raise is that bistability does not guarantee efficiency (which is usually represented by temporal and spatial sparseness of spikes), nor does it guarantee that all weights won’t just end up encoding the same feature. Therefore it is important that the parameters of the Hebbian and homeostasis rules be well-regulated. These kind of fine-tuning can be achieved if we introduce slow homeostatic mechanisms.
Therefore, if we follow the authors’ logic flow of “what mechanisms/ properties must be present to satisfy experimentally observed phenomenons”, we arrive at the conclusion that fast and slow homeostasis mechanisms might have different potential functions: the former in charge of stability, and the latter in charge of fine-tuning parameters.
Written by Belle Liu
Original paper: Zenke, F., & Gerstner, W. (2016). Cooperation across timescales between and Hebbian and homeostatic plasticity.
To begin with, let’s start with a simple form of Hebbian learning, (wij ) ̇=ηxjyi, where wij is the synaptic weight from the jth pre-synaptic neuron to the ith post-synaptic neuron, and xj and yi are the firing rates of the pre- and post-synaptic neurons respectively. Note that this differential equation is inherently unstable, as it is a positive feedback loop with no mechanisms that would either counteract or bound it. Hence this is not biologically plausible (nor useful) at all. It is therefore necessary to introduce other terms that can stabilize it. The two main forms that can do that is 1) introducing a sliding threshold and 2) adding an independent term for rate control. The “threshold” here means that if the firing rate of the neuron is lower than some target value θ, then LTD is induced; if it is higher, then LTP is induced. A “sliding” threshold is therefore one that changes its values according to local variables, and is usually assumed to depend on yi. An example of Hebbian learning plus a sliding threshold is (wij ) ̇=ηxjy(yi-θ(
The rate control term, on the other hand, may take the form of η ̃(κ-(
The next issue even after we accept the premise that one of these forms exist, is that the equation is monostable. Having a monostable system means that it is hard to encode information, since all weights eventually converge to the same attractor. The solution to that is to implement multiple set points, or even a target range. There are many forms of homeostasis that can do this, and the authors of this paper argue that Hebbian plus non-Hebbian plasticity mechanisms, when in the right parameter range, naturally gives rise to bistability.
The final issue that they raise is that bistability does not guarantee efficiency (which is usually represented by temporal and spatial sparseness of spikes), nor does it guarantee that all weights won’t just end up encoding the same feature. Therefore it is important that the parameters of the Hebbian and homeostasis rules be well-regulated. These kind of fine-tuning can be achieved if we introduce slow homeostatic mechanisms.
Therefore, if we follow the authors’ logic flow of “what mechanisms/ properties must be present to satisfy experimentally observed phenomenons”, we arrive at the conclusion that fast and slow homeostasis mechanisms might have different potential functions: the former in charge of stability, and the latter in charge of fine-tuning parameters.
Written by Belle Liu
Original paper: Zenke, F., & Gerstner, W. (2016). Cooperation across timescales between and Hebbian and homeostatic plasticity.
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