Editorial
For most everyday users, mathematics works well enough even if its nature is shrouded in mysteries that are for others to understand. For the clinical scientist, mathematics allows precise conclusions to be reached from fuzzy data—concepts through numbers. For the neurologist, mathematics provides striking examples of how the calculator can go wrong whilst leaving other parts of the cognitive machinery intact: Salomon Henschen (1847–1930) first described ‘acalculia’, concluding that a discrete system independent of the apparatus for orchestrating language and music is needed for calculation even though inability to calculate is often also associated with ‘acopia’ (the inability to copy, not to cope) and ‘number-blindness’. But what is mathematics? In ‘Thoughts of a Mathematician’ Michael Atiyah, reviewing The mathematician's brain by David Ruelle, How mathematicians think by William Byers and The mind of the mathematician by Michael Fitzgerald and Ioan James, explains all (page 1156). Sir Michael is one of the foremost mathematicians of the 20th century: now based in Edinburgh, he previously held professorships in Oxford and Princeton and senior academic positions in Cambridge; a former President of the Royal Society (1990–95) and Master of Trinity College, Cambridge (1990–97), his contributions are in the interaction between geometry, topology and analysis leading to the development of K-theory and the index and fixed point theorems, work for which he has received many civil and academic honours and the highest awards and prizes in mathematics. Striking a reductionist pose, Professor Atiyah offers mathematics as the purest form of thought and one stripped of all external worldly baggage; but he soon reveals his direction of travel by dubbing creative mathematicians as ‘poets of the profession’—albeit basing their muse on hierarchical structures from which the principles of mathematical abstraction naturally emerge. Although, the brain knows the difference between faithful recitation of its 2 x 2 multiplication tables and understanding the significance of numbers, we have to reconcile the debate on whether mathematics is ‘out there’ and just waiting to be found, or the product of a brain-dependent digestion of experience formulated as theorems that help to explain the natural world—cognitive processes that may become engrained, evolved (in a Darwinian sense), hard-wired and, therefore, innate.Michael Atiyah tries various versions of what mathematics is: ‘the study of space and number ... patterns ... order and disorder ... a branch of logic’ but these are all unsatisfactory and represent no more than the scaffolding and foundations of the mathematical edifice. Rather, ‘we mathematicians view ourselves as creative artists, guided by considerations of elegance and beauty’, not to be constrained by rules and most effective when these are broken. He traces in the books under review opinions on the forces that may lead to creativity in mathematics; contradictions and ambiguity (William Byers), personality disorder, notably autism and Asperger's syndrome (Michael Fitzgerald and Ioan James), and idiots savants. But, noting that his mathematical colleagues have omitted any kind of statistical analysis in considering their anecdotal data on abnormal personality and creativity in mathematics, scrutiny of his own acquaintances—simply, all the great mathematicians of the last 50 years—suggests that the mathematician with personality disorder is very much the exception not the rule. His advice is that, if we want to understand cognitive processes as revealed by mathematics, we must unravel the neuroscience of ‘beauty’; what Bertrand Russell (1872–1970) defined as ‘supreme beauty—a beauty cold and austere, like that of sculpture’ (The study of mathematics, 1902) and GH Hardy (1877–1947) considered an ‘imaginary universe so much more beautiful than the stupidly constructed real one’ (A mathematician's apology, 1940).
Six papers in the current issue deal with autism and schizophrenia but not as applied to mathematics. Yong Liu and colleagues writing from Beijing and Changsha (China), Oxford (UK) and Montreal (Canada) map functional arrangements in 90 cortical and subcortical brain regions and show that, over time, many in the prefrontal, parietal and temporal lobes contribute to the loss of connectivity that characterizes schizophrenia (page 945); these alter the efficient ‘small-world’ properties that normally support parallel information transfer through larger and more complex brain networks. Valérian Chambon and a group of investigators from Lyon, Paris and Marseille (France) test an hierarchical model of executive function in which normally we act to exert control over events having a perceptual context and occurring at a specific time (page 962): they show that this interaction breaks down in individuals with schizophrenia through relative inability to choose amongst competing responses that require appreciation of the perceptual component and, hence, the context in which action plans are represented and formulated. Stella Trompet and investigators from Leiden (The Netherlands), Glasgow (UK) and Cork (Ireland) writing on behalf of the PROspective Study of Pravastatin in the Elderly at Risk (PROSPER) Group correlate cognitive function with alleles of the gene for IL-1β converting enzyme (ICE) to show association between the 10643C and 5352A genotypes, reduced IL-1β production and, with some variations in size of effect, improved executive function and memory, thereby identifying potential molecules for neuroprotection from mechanisms of tissue damage that compromise intellectual function (page 1069).
Another molecule recently hitting the neurological headlines is aquaporin 4 (AQP4), mainly as the target for tissue injury and an informative biomarker in neuromyelitis optica. On page 1087, Samira Saadoun and investigators from London (UK) and San Francisco (CA, USA) add to that story the demonstration that, in the experimentally injured spinal cord, AQP4 turns on the tap causing tissue oedema and increasing clinical deficits at follow-up: it follows that inhibition of AQP4 may be an effective neuroprotective strategy for acute spinal cord injury. Efrat Shavit and colleagues from Tel Aviv and Jerusalem (Israel) test the hypothesis that altered nerve conduction involves mediators of the coagulation pathway already known to be activated in inflammatory neuropathy; they demonstrate excess extracellular signal-regulated kinase (ERK) activity in rat sciatic nerve observed following agonist stimulation of protease-activated receptor 1 (PAR-1) and PAR-1 deposition on myelin microvilli at the node of Ranvier that result in reversible conduction block (page 1113). Although clinical features of hippocampal disease are not prominent in multiple sclerosis, the evidence for early cognitive involvement suggests that hippocampal damage is not so rare: on page 1134, Nancy Sicotte and investigators from the University of California (Los Angeles, USA) show selective atrophy of the cornu ammonis 1 (CA1) region in relapsing remitting disease extending to other cornu ammonis regions with the transition to secondary progression, structural changes that correlate with selective impairment of verbal memory encoding and retrieval.
Matti Sillanpää and Dieter Schmidt from Turku (Finland) and Berlin (Germany) report on a cohort of individuals with childhood onset epilepsy studied prospectively for up to 37 years in whom clusters persisting after the introduction of therapy are shown to carry a poor prognosis for overall remission and an increased risk of epilepsy-related death (page 938). Amongst four papers dealing with aspects of visual processing and eye movement, Michael Fahey and colleagues from Melbourne and Sydney (Australia) report reductions in saccadic latency, but not velocity, and abnormalities of ocular fixation and vestibular reflexes correlating with clinical measures of disease severity in Freidreich's ataxia—features that are distinct from other cerebellar ataxias and may serve as biomarkers for Friedreich's ataxia (page 1035). Angela Deutschländer and a team from Munich (Germany) explore the relationship between oscillopsia, retinal slip and visual motion detection to show that patients with unilateral vestibular damage adapt by reducing activation of visual cortical areas for motion-sensitivity and ocular movement—plasticity that serves to suppress the unpleasant experience of oscillopsia (page 1025).
An apocryphal story summarizes the tradition at the National Hospital, Queen Square when clinical neurology was at its zenith: residents at the time considered that the ideal team to handle a case was Sir Charles Symonds (1890–1978) to take the history, Sir Gordon Holmes (1876–1965) to examine the patient and WD Adie (1886–1935) to talk to the family. Perhaps the subtleties of history taking and sensitive explanation are nowhere more expedient than in the evaluation of headache. Elizabeth Cittadini, Manjit Mathuru and Peter Goadsby from London (UK) and San Francisco (USA) describe the clinical features in a series of 31 patients with paroxysmal hemicrania manifesting as short unpleasant episodes with autonomic involvement that are exquisitely sensitive to indometacin; the authors propose re-defining this already well recognized disorder in terms of response to the therapeutic trial of indometacin recommended in every patient experiencing lateralized discrete attacks of head pain associated with autonomic symptoms (page 1142). Classifying headache on the basis of response to treatment is not a new tactic. In From the Archives, we review ‘A particular variety of headache’ by Sir Charles Symonds based on clinical histories carefully elicited in 17 cases observed in consulting practice (Brain 1956: 79; 217–232).
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