Perfect Associative Learning with Spike-Timing-Dependent Plasticity
Recent extensions of the Perceptron as the Tempotron and the Chronotron suggest that this theoretical concept is highly relevant for understanding networks of spiking neurons in the brain. It is not known, however, how the computational power of the Perceptron might be accomplished by the plasticity mechanisms of real synapses. Here we prove that spike-timing-dependent plasticity having an anti-Hebbian form for excitatory synapses as well as a spike-timing-dependent plasticity of Hebbian shape for inhibitory synapses are sufficient for realizing the original Perceptron Learning Rule if these respective plasticity mechanisms act in concert with the hyperpolarisation of the post-synaptic neurons. We also show that with these simple yet biologically realistic dynamics Tempotrons and Chronotrons are learned. The proposed mechanism enables incremental associative learning from a continuous stream of patterns and might therefore underly the acquisition of long term memories in cortex. Our results underline that learning processes in realistic networks of spiking neurons depend crucially on the interactions of synaptic plasticity mechanisms with the dynamics of participating neurons.
The paper presents a learning rule for a simple 
      spiking neuron model based on STDP, where the STDP for excitatory 
      synapses is anti-Hebbian and STDP for inhibitory synapses is Hebbian 
      (i.e., it is reversed in terms of the classical rule). The authors 
      provide a proof that the learning rule implements the Perceptron 
      Learning Rule (PLR) in a specific setup with synchronous input spikes 
      and a kind of teacher forcing. They also show that a modified version 
      of the rule is able to learn spatio-temporal spike patterns to some 
      extend, relating it to tempotron learning. The authors claim that this 
      learning scheme is biologically plausible. General assessment: 
      Quality: The paper addresses an interesting topic. There seems to 
      be a small bug in the theory, but this does not cause a big problem. 
      One problem I see is that the manuscript does not contain a comparison 
      to the simplified tempotron learning rule which is in my opinion at 
      least as biologically plausible as the proposed learning scheme. 
      Another concern is that the simulations for spatio-temporal patterns 
      are done with a modified neuron and plasticity model with some quite 
      ad-hoc modifications. So it is unclear how the initial theoretical 
      investigation corresponds to that. Clarity: The paper is 
      reasonably well written, but clarity could be improved substantially 
      (see below). Originality: The approach is quite probably novel, 
      although there exists some previous work that has not been cited. 
      Significance: The paper seems to be of some relevance. Since the 
      authors do not point to any specific system where this setup could be 
      implemented, I currently do not see any biological evidence for it 
      apart from the fact that anti-Hebbian STDP has been reported. 
      Specific points: 1) Clarity of the presentation 1.1) 
      In line 90, it is stated that neurons are recurrently connected. Yet, I 
      don't' see any place in the paper where this is the case. If I understand 
      correctly, both the theory and the simulations consider simple 
      feed-forward architectures. This is confusing as well as an overstatement 
      and should be corrected. 1.2) It is unclear to me why the authors 
      first introduce the PLR with a {-1,1} coding and then switch to the {0,1} 
      coding. This is just confusing since additional variable-symbols are 
      needed. Just start with the {0,1} coding. I believe that in many 
      text-books it is defined in this way anyways and the mapping between the 
      codings is close to trivial. 1.3) Figure 2 is misleading. The PSPs in 
      Fig. 2 (red traces) are alpha-shaped with a finite rise time. This is 
      misleading since according to the definition the PSPs should really be 
      single exponentials. It is confusing because it does not match the 
      definitions of ~h and ~x in lines 184-186. 1.4) Parameters alpha and 
      beta are used in two different contexts in lines 189 and 310-311. 
      2) Formal proof of equivalence There is one point which maybe 
      I did not understand correctly. I understand that the margin \epsilon 
      for the case of a desired output spike (y_0=1) can be achieved by the 
      after-spike depolarization. This means that during training, because of 
      the teacher spike, the membrane potential has to be somewhat larger than 
      during testing. But I do not see how the analogous case works if the 
      neuron should not spike (y_0=0). In this case, the depolarization needed 
      to elicit a spike during training is exactly the same as during testing. 
      Hence, the neuron cannot learn to bring the depolarization below 
      (threshold-margin). Therefore, using "s" in the right term of the equation 
      in line 204 is probably not correct. It is again misleading that in line 
      198 the authors suggest that beta can be chosen freely. I think this is 
      not the case. I actually see no way to correct for this. The only 
      thing that can be done is to state that the margin for the PLR can only be 
      learned for positive examples (i.e. y_0=1). 3) Comparison to 
      tempotron learning The authors compare their learning rule to 
      tempotron learning in the text, but they do not exhibit an experimental 
      comparison. First, the authors state that the tempotron learning rule is 
      biologically unrealistic (line 55), but this applies only to the full 
      rule, not to the simplified one. In order to see whether the current rule 
      has any advantages over that one, an experimental comparison is needed 
      (possibly also a comparison to the full tempotron rule to see how much one 
      looses). 4) Learning of spatio-temporal patterns 4.1) Several 
      modifications to the original setup are used here which are not well 
      motivated. For example, it is unclear why the hyperpolarization variable 
      nu is introduced in line 284. I suspect it is needed because one wants to 
      avoid that the output neuron spikes more than twice. Also the modification 
      of the learning rule eq. (18) is not clear. It looks like a hack and a 
      assume that the parameters have to be tuned carefully if something changes 
      in the input statistics. I did not understand the explanation in lines 
      300-303. 4.2) Related to the last point. I am not sure whether the 
      initial setup works for spatio-temporal patterns (therefore probably the 
      modifications). The reason is that the second spike of the output neuron 
      acts as a teacher for the following spikes. So even if the desired output 
      is y_0=0, if the neuron erroneously spiked at time t, for input spikes at 
      t'>t, it looks like this was actually a positive pattern. What is the 
      value for "T" in line 320? This value deems critical to me. 5) 
      Relevance The learning setup is that the trained neuron is forced to 
      fire at the same time as the synchronous inputs or before a 
      spatio-temporal input pattern. The former condition (for synchronous 
      input) seems somewhat plausible in some contexts, the latter much less so. 
      In general, if the teacher spike sometimes comes after the input spikes 
      (which seems to be the more plausible situation in vivo) then the system 
      breaks down. Anti-Hebbian STDP has been reported for distal excitatory 
      synapses in pyramidal cells, but on the other hand ref [10] of the 
      manuscript also reported that depolarization would abolish that or make it 
      Hebbian. This would quite probably destroy the properties needed here. 
      Minor points: 6) Literature: The authors may want to 
      compare their rule to ReSume (e.g. Ponulak and Kasinski, Neural 
      Computation 2010) and discuss the relation to the investigations in 
      (Legenstein et al., Neural Computation 2005). 7) It is a bit 
      problematic to define the PLR over gradient descent on eq (7) since this 
      error is not differentiable at E=0. 8) In line x_0^i >= 0 is used, 
      later x_0^i \in {0,1}. 9) line 360: "biologically highly plausible" is 
      an overstatement. 10) line 367: "substantial capacity". This is not 
      clear. What is a substantial capacity in this case? Comparison with 
      tempotron capacity etc.