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 nonEuclidean spaces
nonlinear dynamics
isodipole textures
point processes
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Last revised: 4/2/11
Spike metrics
brief background
key reference
further reading

Develop a computationally efficient algorithm for
D^{motif}.
Maybe this is a wellknown problem in dynamic programming algorithms.

Develop a computationally efficient algorithm for
combinations
of
D^{spike},
D^{interval},
and D^{motif}.
Maybe this is also a wellknown problem in dynamic programming algorithms.
Progress (8/2002) for combinations of D^{spike} and D^{interval}:
Perhaps any optimal path from spike train A to spike train B can be reorganized as an equalcost path from spike train A to spike train X,
then from X to B, in which A to X
only uses the transformations of D^{spike}, and X to B only uses the transformations of D^{interval}.
Then, find a DP algorithm that finds X. Unfortunately, such reorganizations cannot be guaranteed.
This idea makes use of the "algebraic" properties of the transformations. For general inspiration, see the algebraic dynamic programming work of
Robert Giegerich.

Develop a taxonomy of spike metrics which is "complete" in
some biologicallyreasonable sense. The topological hierarchy is
interesting, but even topologically equivalent spike metrics
must be distinguished.

Understand the relationship of the spike metrics to
Euclidean distances.
One line of attack
Consider a monotonic transformation F(x) applied to the metrics.
For F(x)=x^{p} (0 < p < 1),
F(D(.,.)) is also a metric, with the same
topology, Voronoi neighborhoods, and rankorder
as D(.,.).
I had hypothesized that for p = 1/2, one can always achieve an embedding with Euclidean geometry. (It is easy to show that p <= 1/2 is required for large q, and that p = 1 suffices for q = 0.)
p = 1/2 also seemed plausible because a lattice of points in the Minkowski city block space  which appears to be related to the spike time metric 
can be embedded in a Euclidean space with F(x)=x^{1/2}, in a manner that preserves the metric. (For a kdimensional lattice of N_{1}N_{2}...N_{k} points, then the dimension of the Euclidean space need be no larger than N_{1}+N_{2}+...+N_{k}k. My interest is that the embedding requires a uniform power law transformation, independent of the number of points. This is not always possible for a noneuclidean manifold. The more general question of when a nonEuclidean manifold can be embedded in a distancepreserving (up to a power law transformation) manner in a Euclidean space, is an interesting one  see multidimensional scaling of symmetric nonEuclidean spaces below.)
But it is clear, due to the work of
Dmitriy Aronov,
that no exponent p>0
can guarantee a Euclidean embedding of a set of spike trains under the spike time metric.
See the paper.
Thus it would seem that no universal scalefree transformation of the distance suffices to embed
the space of the spike time distance into Euclidean space.
Another line of attack
Consider two spike trains s_{1} and s_{2}
generated by timevarying Poisson processes,
with rates r_{1}(t) and r_{2}(t). Can one estimate (perhaps, asymptotically
in the limit of high firing rates) D^{spike}[q](s_{1},s_{2}), and then
relate this to the squared Euclidean distance between the rate functions
r_{1}(t) and r_{2}(t)? Intuitively, this relationship should approach
a bilinear function when rates are high, similar to smoothing of the
rate functions r_{1}(t) and r_{2}(t) by a kernel whose support is
[1/q,1/q].
Is this true?
If such a bilinear function can be found, what are its eigenvalues and eigenvectors?
 Develop an esthetically acceptable extension of
the "direct" method for the calculation of information
(de Ruyter van Steveninck, R. R., Lewen, G.D., Strong, S.P., Koberle, R., & Bialek, W.
(1997) Reproducibility and variability in neural spike trains. Science 275, 18051808)
that encompasses the metric approach. The "direct" method
is based on comparison of the volumes that sets of spike trains
occupy in a vector space. Can quantities that serve as these volumes be estimated
if only metrical data are available?
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Isodipole textures and related
top
brief background
review article
further reading
examples
 What are the
scaleinvariant limits of isodipole textures
constructed from
plane Markov processes
other than
the standard even and odd textures?
The thirdorder
triangle textures are a good place to start.
 What are the scaleinvariant limits
of planar textures in general?
 What are the statistics of induced correlations
(i.e., longerrange correlations forced by the defining recursion rule)
of
isodipole textures
constructed from
plane Markov processes
other than
the standard even, odd, and triangle textures?
For 2x2 cliques and binary textures, these textures have been fully parameterized.
F. Champagnat, J. Idier, and Y. Goussard, Stationary Markov random fields on a rectangular finite lattice, IEEE Trans. Inf. Theory 44, 29012916 (1998), but this parameterization is awkward, and the general question remains open.
See this paper for a psychophysical exploration of a 2parameter family of maximum entropy textures contained within this parameterization.

What are efficient ways to sample maximumentropy distributions constrained by local correlations, when the local correlations are of order >=3 and/or the Pickard rules are violated? Many special cases have been solved
(Victor, J.D., Thengone, D., and Conte, M.M.: Analyzing local image statistics via maximumentropy constructions, CoSyNe 2011 abstract), but the general problem is open.
Multidimensional scaling of nonEuclidean spaces
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It is well known that a Riemannian manifold can be embedded in a higherdimensional Euclidean space in a topologypreserving 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 scalefree (as discussed above), we require F(x)=x^{p}. 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.
 City block distance: p=1/2 suffices, as described above.
 Other Minkowski spaces: not settled
 Lie groups, such as SO(n): demonstrated NOT possible for any p>0, for SO(3) and above
 The surface of an nsphere: p=1/2 suffices, up to a 5sphere (2sphere= ordinary sphere), and appears that p=1/2 suffices for any nsphere  extensive numerical evidence; proof will follow from a series that needs to be summed. These results follow from an application of the group representation theory for SO(n), and lead to some interesting series for pi and relationships involving the Catalan numbers. Work in progress.
Nonlinear autoregressive models
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brief background
key reference
further reading
 What is a systematic way for determining whether a
model dynamical system (e.g., defined by a set of differential
equations) is consistent with the qualitative features of an NLAR
fingerprint?
More generally, what is the relationship between the NLAR fingerprint
and global dynamics?
 What is a sensible way (for multichannel data)
to merge principal components analysis and
NLAR fingerprints?
Empirically, in a multichannel EEG dataset,
it often appears that only one or two
principal components contain the bulk of the nonlinear contributions.
However, attempting to isolate the nonlinear contributions
in a single source via "independent components analysis" fails,
even for simple model systems, whenever the nonlinearity is strong
(consider one channel given by the square of another).
Point processes
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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 crosscorrelation 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:
 Create three point processes, X, Y, and Z. Let neuron A's spike train be the union of X and Y, and let neuron B's spike train be the union of Y and Z. IF X, Y and Z are all Poisson, then so are A and B. The spikes in Y lead to positive correlation at 0 lag.
 (Thanks to Bruce Knight) Consider a point process in which the intervals are alternately and independently taken from a gamma process of order g_{1} and a gamma process of order g_{2}, where g_{1}+g_{2}=1. Assign the evennumbered spikes are assigned to neuron A and the oddnumbered spikes to neuron B. These two spike trains are Poisson (since the convolution of two gamma distributions is another gamma distribution of order equal to the sum of the original orders). A and B are positively correlated for short lags.
But neither of these approaches readily generalizes to a procedure to create negative correlations.
Solution!
(3/2001) Daniel Fisher.
His solution (which is a generalization of the second strategy above) is given here:
The AtoB point process has
interval distribution
S=y[(a1)^{2}e^{t} + ((g1)^{2}(a1)^{2})e^{gt}  (g1)(ga)^{2}t e^{gt}] with y=(g/a(g1))^{2}
and the BtoA point process has interval distribution
R=z[delta(t) + (2(ga) + (ga)^{2}t)e^{at}] with z=(a/g)^{2}.
Take g > a > 1 with ln(ga)  ln(a1) < 1+ (a1)/(ga).
These conditions ensure that both R(t) and S(t) are positive and that the cross correlation (with mean subtracted)
has a minimum value
 (11/a) e^{2/(ga)}
that is negative. By choosing both g and a large, one can get this minimum arbitrarily close to
1. With these forms, the crosscorrelation 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, 238255 (1979)
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