Capacity of a POST Channel with and without Feedback

We consider finite state channels where the state of the channel is its previous output. We refer to these as POST (Previous Output is the STate) channels. We first focus on POST($\alpha$) channels. These channels have binary inputs and outputs, where the state determines if the channel behaves as a $Z$ or an $S$ channel, both with parameter $\alpha$. %with parameter $\alpha.$ We show that the non feedback capacity of the POST($\alpha$) channel equals its feedback capacity, despite the memory of the channel. The proof of this surprising result is based on showing that the induced output distribution, when maximizing the directed information in the presence of feedback, can also be achieved by an input distribution that does not utilize of the feedback. We show that this is a sufficient condition for the feedback capacity to equal the non feedback capacity for any finite state channel. We show that the result carries over from the POST($\alpha$) channel to a binary POST channel where the previous output determines whether the current channel will be binary with parameters $(a,b)$ or $(b,a)$. Finally, we show that, in general, feedback may increase the capacity of a POST channel

Finite-State Channels with Feedback and State Known at the Encoder

We consider finite state channels (FSCs) with feedback and state information known causally at the encoder. This setting is quite general and includes: a memoryless channel with i.i.d. state (the Shannon strategy), Markovian states that include look-ahead (LA) access to the state and energy harvesting. We characterize the feedback capacity of the general setting as the directed information between auxiliary random variables with memory to the channel outputs. We also propose two methods for computing the feedback capacity: (i) formulating an infinite-horizon average-reward dynamic program; and (ii) a single-letter lower bound based on auxiliary directed graphs called $Q$-graphs. We demonstrate our computation methods on several examples. In the first example, we introduce a channel with LA and derive a closed-form, analytic lower bound on its feedback capacity. Furthermore, we show that the mentioned methods achieve the feedback capacity of known unifilar FSCs such as the trapdoor channel, the Ising channel and the input-constrained erasure channel. Finally, we analyze the feedback capacity of a channel whose state is stochastically dependent on the input.Comment: 39 pages, 10 figures. The material in this paper was presented in part at the 56th Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, USA, October 2018, and at the IEEE International Symposium on Information Theory, Los Angeles, CA, USA, June 202

Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review