Unsolved mathematical problems

Here are some unsolved mathematical problems with potential impact for neuroscience. Not all are tightly posed. Any feedback, not limited to brilliant ideas, is always appreciated and will be gratefully acknowledged.

spike metrics
multidimensional scaling of symmetric non-Euclidean spaces
nonlinear dynamics
isodipole textures
point processes
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Last revised: 4/2/11

Spike metrics

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Isodipole textures and related

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Multidimensional scaling of non-Euclidean spaces


It is well known that a Riemannian manifold can be embedded in a higher-dimensional Euclidean space in a topology-preserving manner. Here we ask about whether it is possible to find an embedding of a Riemannian manifold that preserves, or almost preserves, distance. That is, the geodesic distance between two points a and b, d(a,b), is required to be equal to some function of the Euclidean distance between the embedded images z(a) and z(b) of the points, namely, d(a,b)=F(|z(a)-z(b)|). So that the transformation is scale-free (as discussed above), we require F(x)=xp. We are interested in finding a power p that is independent of the number of points within the manifold to be embedded, an are not concerned whether the number of dimensions is finite or infinite.

Nonlinear autoregressive models

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Point processes


Consider: (i) The Poisson process is often taken to be a good starting place for modeling spike trains. (ii) Spike trains from functionally related neurons are often positively or negatively correlated at short lags. But is it possible to construct a pair of point processes, each of which is Poisson when viewed in isolation, whose cross-correlation is negative at short lags? If it is impossible, can this be proven?

Note that it is easy to construct Poisson processes that are positively correlated at short lags. Two strategies:

But neither of these approaches readily generalizes to a procedure to create negative correlations.


(3/2001) Daniel Fisher. His solution (which is a generalization of the second strategy above) is given here: The A-to-B point process has interval distribution
S=y[(a-1)2e-t + ((g-1)2-(a-1)2)e-gt - (g-1)(g-a)2t e-gt] with y=(g/a(g-1))2

and the B-to-A point process has interval distribution
R=z[delta(t) + (2(g-a) + (g-a)2t)e-at] with z=(a/g)2.

Take g > a > 1 with ln(g-a) - ln(a-1) < 1+ (a-1)/(g-a).
These conditions ensure that both R(t) and S(t) are positive and that the cross correlation (with mean subtracted) has a minimum value
- (1-1/a) e-2/(g-a)
that is negative. By choosing both g and a large, one can get this minimum arbitrarily close to -1. With these forms, the cross-correlation has a delta function component at t=0, is large for small t, dips below zero for larger t and asymptotically approaches zero from below for large t.

(3/15/07) An extensive analysis of these constructions, and of alternating Poisson processes in general, has been posted by Don Johnson. Download it.

Related reference

Contributed (8/12/2002) by Dario Ringach: Griffiths et al., "Aspects of correlation in bivariate Poisson distributions and processes", Aust. J. Statist. 21, 238-255 (1979)


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