skip to main content


Title: A dynamic spike threshold with correlated noise predicts observed patterns of negative interval correlations in neuronal spike trains
Abstract

Negative correlations in the sequential evolution of interspike intervals (ISIs) are a signature of memory in neuronal spike-trains. They provide coding benefits including firing-rate stabilization, improved detectability of weak sensory signals, and enhanced transmission of information by improving signal-to-noise ratio. Primary electrosensory afferent spike-trains in weakly electric fish fall into two categories based on the pattern of ISI correlations: non-bursting units have negative correlations which remain negative but decay to zero with increasing lags (Type I ISI correlations), and bursting units have oscillatory (alternating sign) correlation which damp to zero with increasing lags (Type II ISI correlations). Here, we predict and match observed ISI correlations in these afferents using a stochastic dynamic threshold model. We determine the ISI correlation function as a function of an arbitrary discrete noise correlation function$${{\,\mathrm{\mathbf {R}}\,}}_k$$Rk, wherekis a multiple of the mean ISI. The function permits forward and inverse calculations of the correlation function. Both types of correlation functions can be generated by adding colored noise to the spike threshold with Type I correlations generated with slow noise and Type II correlations generated with fast noise. A first-order autoregressive (AR) process with a single parameter is sufficient to predict and accurately match both types of afferent ISI correlation functions, with the type being determined by the sign of the AR parameter. The predicted and experimentally observed correlations are in geometric progression. The theory predicts that the limiting sum of ISI correlations is$$-0.5$$-0.5yielding a perfect DC-block in the power spectrum of the spike train. Observed ISI correlations from afferents have a limiting sum that is slightly larger at$$-0.475 \pm 0.04$$-0.475±0.04($$\text {mean} \pm \text {s.d.}$$mean±s.d.). We conclude that the underlying process for generating ISIs may be a simple combination of low-order AR and moving average processes and discuss the results from the perspective of optimal coding.

 
more » « less
NSF-PAR ID:
10375779
Author(s) / Creator(s):
; ; ;
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Biological Cybernetics
Volume:
116
Issue:
5-6
ISSN:
1432-0770
Page Range / eLocation ID:
p. 611-633
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    Environmental seismic disturbances limit the sensitivity of LIGO gravitational wave detectors. Trains near the LIGO Livingston detector produce low frequency (0.5–10Hz) ground noise that couples into the gravitational wave sensitive frequency band (10–100Hz) through light reflected in mirrors and other surfaces. We investigate the effect of trains during the Advanced LIGO third observing run, and propose a method to search for narrow band seismic frequencies responsible for contributing to increases in scattered light. Through the use of the linear regression tool Lasso (least absolute shrinkage and selection operator) and glitch correlations, we identify the most common seismic frequencies that correlate with increases in detector noise as 0.6–0.8Hz, 1.7–1.9Hz, 1.8–2.0Hz, and 2.3–2.5Hzin the LIGO Livingston corner station.

     
    more » « less
  2. Abstract

    It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$Lβ,γ=-divDd+1+γ-nassociated to a domain$$\Omega \subset {\mathbb {R}}^n$$ΩRnwith a uniformly rectifiable boundary$$\Gamma $$Γof dimension$$d < n-1$$d<n-1, the now usual distance to the boundary$$D = D_\beta $$D=Dβgiven by$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$Dβ(X)-β=Γ|X-y|-d-βdσ(y)for$$X \in \Omega $$XΩ, where$$\beta >0$$β>0and$$\gamma \in (-1,1)$$γ(-1,1). In this paper we show that the Green functionGfor$$L_{\beta ,\gamma }$$Lβ,γ, with pole at infinity, is well approximated by multiples of$$D^{1-\gamma }$$D1-γ, in the sense that the function$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$|D(ln(GD1-γ))|2satisfies a Carleson measure estimate on$$\Omega $$Ω. We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).

     
    more » « less
  3. Abstract

    The azimuthal ($$\Delta \varphi $$Δφ) correlation distributions between heavy-flavor decay electrons and associated charged particles are measured in pp and p–Pb collisions at$$\sqrt{s_{\mathrm{{NN}}}} = 5.02$$sNN=5.02TeV. Results are reported for electrons with transverse momentum$$44<pT<16$$\textrm{GeV}/c$$GeV/c and pseudorapidity$$|\eta |<0.6$$|η|<0.6. The associated charged particles are selected with transverse momentum$$11<pT<7$$\textrm{GeV}/c$$GeV/c, and relative pseudorapidity separation with the leading electron$$|\Delta \eta | < 1$$|Δη|<1. The correlation measurements are performed to study and characterize the fragmentation and hadronization of heavy quarks. The correlation structures are fitted with a constant and two von Mises functions to obtain the baseline and the near- and away-side peaks, respectively. The results from p–Pb collisions are compared with those from pp collisions to study the effects of cold nuclear matter. In the measured trigger electron and associated particle kinematic regions, the two collision systems give consistent results. The$$\Delta \varphi $$Δφdistribution and the peak observables in pp and p–Pb collisions are compared with calculations from various Monte Carlo event generators.

     
    more » « less
  4. Abstract

    In a Merlin–Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability 1, and rejects invalid proofs with probability arbitrarily close to 1. The running time of such a system is defined to be the length of Merlin’s proof plus the running time of Arthur. We provide new Merlin–Arthur proof systems for some key problems in fine-grained complexity. In several cases our proof systems have optimal running time. Our main results include:

    Certifying that a list ofnintegers has no 3-SUM solution can be done in Merlin–Arthur time$$\tilde{O}(n)$$O~(n). Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in$$\tilde{O}(n^{1.5})$$O~(n1.5)time (that is, there is a proof system with proofs of length$$\tilde{O}(n^{1.5})$$O~(n1.5)and a deterministic verifier running in$$\tilde{O}(n^{1.5})$$O~(n1.5)time).

    Counting the number ofk-cliques with total edge weight equal to zero in ann-node graph can be done in Merlin–Arthur time$${\tilde{O}}(n^{\lceil k/2\rceil })$$O~(nk/2)(where$$k\ge 3$$k3). For oddk, this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in anm-edge graph can be done in Merlin–Arthur time$${\tilde{O}}(m)$$O~(m). Previous Merlin–Arthur protocols by Williams [CCC’16] and Björklund and Kaski [PODC’16] could only countk-cliques in unweighted graphs, and had worse running times for smallk.

    Computing the All-Pairs Shortest Distances matrix for ann-node graph can be done in Merlin–Arthur time$$\tilde{O}(n^2)$$O~(n2). Note this is optimal, as the matrix can have$$\Omega (n^2)$$Ω(n2)nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an$$\tilde{O}(n^{2.94})$$O~(n2.94)nondeterministic time algorithm.

    Certifying that ann-variablek-CNF is unsatisfiable can be done in Merlin–Arthur time$$2^{n/2 - n/O(k)}$$2n/2-n/O(k). We also observe an algebrization barrier for the previous$$2^{n/2}\cdot \textrm{poly}(n)$$2n/2·poly(n)-time Merlin–Arthur protocol of R. Williams [CCC’16] for$$\#$$#SAT: in particular, his protocol algebrizes, and we observe there is no algebrizing protocol fork-UNSAT running in$$2^{n/2}/n^{\omega (1)}$$2n/2/nω(1)time. Therefore we have to exploit non-algebrizing properties to obtain our new protocol.

    Certifying a Quantified Boolean Formula is true can be done in Merlin–Arthur time$$2^{4n/5}\cdot \textrm{poly}(n)$$24n/5·poly(n). Previously, the only nontrivial result known along these lines was an Arthur–Merlin–Arthur protocol (where Merlin’s proof depends on some of Arthur’s coins) running in$$2^{2n/3}\cdot \textrm{poly}(n)$$22n/3·poly(n)time.

    Due to the centrality of these problems in fine-grained complexity, our results have consequences for many other problems of interest. For example, our work implies that certifying there is no Subset Sum solution tonintegers can be done in Merlin–Arthur time$$2^{n/3}\cdot \textrm{poly}(n)$$2n/3·poly(n), improving on the previous best protocol by Nederlof [IPL 2017] which took$$2^{0.49991n}\cdot \textrm{poly}(n)$$20.49991n·poly(n)time.

     
    more » « less
  5. Abstract

    We prove that the Hilbert scheme ofkpoints on$${\mathbb {C}}^2$$C2($$\hbox {Hilb}^k[{\mathbb {C}}^2]$$Hilbk[C2]) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$${\mathbb {C}}^\times _\hbar $$Cħ×-action. First, we find a two-parameter family$$X_{k,l}$$Xk,lof self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$Hilbk[C2]is obtained via direct limit$$l\longrightarrow \infty $$land by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$\hbar $$ħ-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-Nsheaves on$${\mathbb {P}}^2$$P2with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.

     
    more » « less