At present there is no generally accepted theory of how cognitive phenomena arise from computations in cortex. Further, there is no consensus on how the search for one should be refocussed so as to make it more fruitful. In this short piece we observe that research in computer science over the last several decades has shown that significant computational phenomena need to circumvent significant inherent quantitative impediments, such as of computational complexity. We argue that computational neuroscience has to be informed by the same quantitative concerns for it to succeed. It is conceivable that the brain is the one computation that does not need to circumvent any such obstacles, but if that were the case then quantitatively plausible theories of cortex would now surely abound and be driving experimental investigations. Introduction That computing is the right framework for understanding the brain became clear to many soon after the discovery of universal computing by Turing [1], who was himself motivated by the question of understanding the scope of human mental activity. McCulloch and Pitts [2] made a first attempt to formalize neural computation, pointing out that their networks were of equivalent expressive power to Turing machines. By the 1950s it was widely recognized that any science of cognition would have to be based on computation. It would probably come as a shock to the earliest pioneers, were they to return today, that more progress has not been made towards a generally agreed computational theory of cortex. They may have expected, short of such a generally agreed theory, that today there would at least exist a variety of viable competing theories. Understanding cortex is surely among the most important questions ever posed by science. Astonishingly, the question of proposing general theories of cortex and subjecting them to experimental examination is currently not even a mainstream scientific activity. Our review here is informed by the observation that since Marr's time computer science has made very substantial progress in certain quantitative directions. The following four phenomena are clearly critical for the brain: communication, computation, learning and evolution. Over the last few decades all four have been subject to quantitative analysis, and are now known to be subject to hard quantitative constraints (see We do not believe that there can be any doubt that the theory sought has to be computational in the general sense of Turing. The question that arises is: In what way does Marr's articulation of the computational approach fall short? Our answer is that, exactly as in any other domains of computation, a successful theory will have to show additionally, how the quantitative challenges that need to be faced are solved in cortex. If these challenges were nonexistent or insignificant then plausible theories would now abound and the only task remaining for us would be to establish which one nature is using. This augmented set of requirements is quite complex in that many issues have to be faced simultaneously. We suggest the following as a streamlined working formulation for the present: (i) Specify a candidate set of quantitatively challenging cognitive tasks that cortex may be using as the primitives from which it builds cognition. At a minimum, this set has to include the task of memorization, and some additional tasks that use the memories created. The task set needs to encompass both the learning and the execution of the capabilities in question. (ii) Explain how, on a model of computation that faithfully reflects the quantitative resources that cortex has available, instances of these tasks can be realized by explicit algorithms. (iii) Provide some plausible experimental approach to confirming or falsifying the theory as it applies to cortex. (iv) Explain how there may be an evolutionary path to the brain having acquired these capabilities. To illustrate that this complex of requirements can be pursued systematically together we shall briefly describe the framework developed for this by the author Positive representations In order to specify computational tasks in terms of inputoutput behavior one needs to start with a representation for each task. It is necessary to ensure that for any pair of tasks where the input of one is the output of the other there is a common representation at that interface. Here we shall take the convenient course of having a common representation for all the tasks that will be considered, so that their composability will follow. In a positive representation [5] a real world item (a concept, event, individual, etc.) is represented by a set S of r neurons. A concept being processed corresponds to the members of S firing in a distinct way. More precisely, as elaborated further in Positive representations come in two varieties, disjoint, which means that the S's of distinct concepts are disjoint, and shared, which means that the S's can share neurons. Disjointness makes computation easier but requires small r (such as r = 50) if large numbers of concepts are to be represented. The shared representation allows for more concepts to be represented (especially necessary if r is very large, such as several percent of the total number of neurons) but can be expected to make computation, without interference among the task instances, more challenging. Random access versus local tasks We believe that cortex is communication bounded in the sense that: (i) each neuron is connected to a minute fraction of all the other neurons, (ii) each individual synapse typically has weak influence, in that a presynaptic action potential will make only a small contribution to the threshold potential needed to be overcome in the postsynaptic cell, and (iii) there is no global addressing mechanism as computers have. We call tasks that potentially require communication between arbitrary memorized concepts random-access tasks. Such tasks, for example, an association between an arbitrary pair of concepts, are the most demanding in communication and therefore quantitatively the most challenging for the brain to realize. The arbitrary knowledge structures in the world will have to be mapped, by the execution of a sequence of random access tasks that only change synaptic weights, to the available connections among the neurons that are largely fixed at birth. We distinguish between two categories of tasks. Tasks from the first category assign neurons to a new item. We have just one task of this type, which we call Hierarchical Memorization and define it as follows: For any stored items A, B, allocate neurons to new item C and make appropriate changes in the circuit so that in future A and B active will cause C to be active also. The second category of tasks make modifications to the circuits so as to relate in a new way items to which neurons have been already assigned. We consider the following three. Association: For any stored items A, B, change the www.sciencedirect.com circuit so that in future when A is active then B will be caused to be also. Supervised Memorization of Conjunctions: For stored items A, B, C change the circuits so that in future A and B active will cause C to be active also. Inductive Learning of Simple Threshold Functions: for one stored item A learn a criterion in terms of the others. This third operation is the one that achieves generalization, in that appropriate performance even on inputs never before seen is expected. The intention is that any new item to be stored will be stored in the first instance as a conjunction of items previously memorized (which may be visual, auditory, conceptual, etc.) Once an item has neurons allocated, it becomes an equal citizen with items previously stored in its ability to become a constituent in future actions. These actions can be the creation of further concepts using the hierarchical memorization operation, or establishing relationships among the items stored using one of the operations of the second kind, such as association. The latter operations can be regarded as the workhorses of the cognitive system, building up complex data structures reflecting the relations that exist in the world among the items represented. However, each such operation requires each item it touches to have been allocated in the first instance by a task of the first kind. Random access tasks are the most appropriate for our study here since, almost by definition, they are the most challenging for any communication bound system. For tasks that require only local communication, such as aspects of low-level vision, viable computational solutions may be more numerous, and quantitative studies may be less helpful in identifying the one nature has chosen. We emphasize that for the candidate set it is desirable to target from the start a mixed set of different task types as here, since such sets are more likely to form a sufficient set of primitives for cognition. Previous approaches have often focused on a single task type The neuroidal model Experience in computer science suggests that models of computation need to be chosen carefully to fit the problem at hand. The criterion of success is the ultimate usefulness of the model in illuminating the relevant phenomena. In neuroscience we will, no doubt, ultimately need a variety of models at different levels. The neuroidal model is designed to explicate phenomena around the random access tasks we have described, where the constraints are dictated by the gross communication constraints on cortex rather than the detailed computations inside neurons. The neuroidal model has three main numerical parameters: n, the number of neurons, d the number of connections per neuron, and k, the minimum number of presynaptic neurons needed to cause an action potential in a postsynaptic neuron (in other words the maximum synaptic strength is 1/k times the neuron threshold). Each neuron can be in one of a finite number of states and each synapse has some strength. These states and strengths are updated according to purely local rules using computationally weak steps. Each update will be influenced by the firing pattern of the presynaptic neurons according to a function that is symmetric in those inputs. There is a weak timing mechanism that allows the neurons to count time accurately enough so stay synchronized with other neurons for a few steps
