Paper:
A memory frontier for complex synapses
An incredible gulf separates theoretical models of synapses, often described solely by a single scalar value denoting the size of a postsynaptic potential, from the immense complexity of molecular signaling pathways underlying real synapses. To understand the functional contribution of such molecular complexity to learning and memory, it is essential to expand our theoretical conception of a synapse from a single scalar to an entire dynamical system with many internal molecular functional states. Moreover, theoretical considerations alone demand such an expansion; network models with scalar synapses assuming finite numbers of distinguishable synaptic strengths have strikingly limited memory capacity. This raises the fundamental question, how does synaptic complexity give rise to memory? To address this, we develop new mathematical theorems elucidating the relationship between the structural organization and memory properties of complex synapses that are themselves molecular networks. Moreover, in proving such theorems, we uncover a framework, based on first passage time theory, to impose an order on the internal states of complex synaptic models, thereby simplifying the relationship between synaptic structure and function.

Review:
The paper studies the problem of memory storage with 
      discrete (digital) synapses. Previous work established that memory 
      capacity can be increased by adding a cascade of (latent) states but the 
      optimal state transition dynamics was unknown and the actual dynamics was 
      usually hand-picked using some heuristic rules. In this paper the authors 
      aim to derive the optimal transition dynamics for synaptic cascades. They 
      first derive an upper bound on achievable memory capacity and show that 
      simple models with linear chain structures can approach (achieve) this 
      bound. The paper is clear, high quality, generally well written 
      and has a clear contribution to the field. Minor comments: 
      line 195: Different systems in the brain might be optimised for 
      storage at different timescale. Are there any evidence for that? line 
      220: When looking at the molecular network of a synapse it is not 
      immediately obvious what are the "states" of the system and what is the 
      associated state transition dynamics. A strong prediction of the paper is 
      that the molecular network in the synapse has a meaningful structure for 
      information storage, not like the one at Fig 2a-b. Is there a simple way 
      to check this prediction? line 294: The authors may refer to the 
      supplementary material around equation 14. line 299: Isn't it 
      problematic, than nu_i depends on M - that we try to optimize? line 
      304: An intuitive description of the process depicted on Fig. 3a-b is that 
      a potentiation has to move all synaptic states to more potentiated states, 
      while a depression event must depress it. line 326: In all the 
      examples provided in the paper, half of the states are associated with 
      w=-1 the other half has w=+1. How the memory capacity changes if the 
      states are more asymmetric, e.g., in the most extreme case there is only a 
      single state with w=-1 and M-1 state with w=1? line 415: Linear chains 
      maximize area - but not necessarily achieve the best performance (touches 
      the envelope) at t_0? Is it true that for high performance at a certain 
      time a chain other than linear may be optimal? Is the chain shown on Fig5b 
      a linear chain? I appreciate the detailed rebuttal from the 
      authors. A short definition of the linear chain would be more useful than 
      repeating Fig 4. A possible definition would be that a chain is linear if 
      there is a unique ordering of the nodes from depressed to potentiated - 
      but this is not what the authors used.