📄 node26.html
字号:
<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.0//EN"><!--Converted with LaTeX2HTML 96.1-h (September 30, 1996) by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds --><HTML><HEAD><TITLE>Spike trains</TITLE><META NAME="description" CONTENT="Spike trains"><META NAME="keywords" CONTENT="Surrogates"><META NAME="resource-type" CONTENT="document"><META NAME="distribution" CONTENT="global"><LINK REL=STYLESHEET HREF="Surrogates.css"></HEAD><BODY bgcolor=#ffffff LANG="EN" > <A NAME="tex2html335" HREF="node27.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html333" HREF="node22.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html329" HREF="node25.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html336" HREF="node27.html">Questions of interpretation</A><B>Up:</B> <A NAME="tex2html334" HREF="node22.html">Various Examples</A><B> Previous:</B> <A NAME="tex2html330" HREF="node25.html">Uneven sampling</A><BR> <P><H2><A NAME="SECTION00064000000000000000">Spike trains</A></H2><P><P>A spike train is a sequence of <I>N</I> events (for example neuronal spikes, orheart beats) occurring at times <IMG WIDTH=28 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2310" SRC="img132.gif">. Variations in the events beyond theirtiming are ignored. Let us first note that this very common kind of data isfundamentally different from the case of unevenly <EM>sampled</EM> time seriesstudied in the last section in that the sampling instances <IMG WIDTH=28 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2310" SRC="img132.gif"> are notindependent of the measured process. In fact, between these instances, thevalue of <I>s</I>(<I>t</I>) is undefined and the <IMG WIDTH=28 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2310" SRC="img132.gif"> contain all the informationthere is.<P>Very often, the discrete sequence if inter-event intervals <IMG WIDTH=101 HEIGHT=18 ALIGN=MIDDLE ALT="tex2html_wrap_inline2380" SRC="img163.gif"> istreated as if it were an ordinary time series. We must keep in mind, however,that the index <I>n</I> is not proportional to time any more. It depends on thenature of the process if it is more reasonable to look for correlations in timeor in event number. Since in the latter case we can use the standard machineryof regularly sampled time series, let us concentrate on the more difficult realtime correlations.<P><blockquote><A NAME="972"> </A><IMG WIDTH=357 HEIGHT=147 ALIGN=BOTTOM ALT="figure1086" SRC="img160.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figrrseries"> </A> Heart rate fluctuations seen by plotting the time interval between consecutive heart beats (R waves) versus the beat number. Note that the spread of values is rather small due to the near-periodicity of the heart beat. <BR></blockquote><P><blockquote><A NAME="974"> </A><IMG WIDTH=352 HEIGHT=253 ALIGN=BOTTOM ALT="figure1087" SRC="img162.gif"><BR><STRONG>Figure:</STRONG> <A NAME="figrr"> </A> Binned autocorrelation function of an RR interval time series. Upper panel: <IMG WIDTH=31 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2060" SRC="img57.gif"> and <IMG WIDTH=32 HEIGHT=29 ALIGN=MIDDLE ALT="tex2html_wrap_inline2368" SRC="img161.gif"> are practically indistinguishable. Lower: Autocorrelation for a random scramble of the data. Note that most of the periodicity is given by the fact that the duration of each beat had rather little variation during this recording.<BR></blockquote><P>In particular the literature on <EM>heart rate variability</EM> (HRV) containsinteresting material on the question of spectral estimation and linear modelingof spike trains, here usually inter-beat (RR) interval series, seee.g. Ref. [<A HREF="node36.html#spikespec">51</A>]. For the heart beat interval sequence shown inFig. <A HREF="node26.html#figrrseries">16</A>, spectral analysis of <IMG WIDTH=17 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline2384" SRC="img164.gif"> versus <I>n</I> may revealinteresting structure, but even the mean periodicity of the heart beat would belost and serious aliasing effects would have to be faced. A very convenientand powerful approach that uses the real time <I>t</I> rather than the event number<I>n</I> is to write a spike train as a sum of Dirac delta functions located at thespike instances:<BR><A NAME="eqdelta"> </A><IMG WIDTH=500 HEIGHT=46 ALIGN=BOTTOM ALT="equation1088" SRC="img165.gif"><BR>With <IMG WIDTH=186 HEIGHT=33 ALIGN=MIDDLE ALT="tex2html_wrap_inline2392" SRC="img166.gif">, theperiodogram spectral estimator is then simply obtained by squaring the(continuous) Fourier transform of <I>s</I>(<I>t</I>):<BR><A NAME="eqspower"> </A><IMG WIDTH=500 HEIGHT=53 ALIGN=BOTTOM ALT="equation1090" SRC="img167.gif"><BR>Other spectral estimators can be derived by smoothing <IMG WIDTH=34 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2396" SRC="img168.gif"> or by datawindowing. It is possible to generate surrogate spike trains that realise thespectral estimator Eq.(<A HREF="node26.html#eqspower">28</A>), although this is computationally verycumbersome. Again, we can take advantage of the relative computational ease ofbinned autocorrelations here.<A NAME="tex2html21" HREF="footnode.html#539"><IMG ALIGN=BOTTOM ALT="gif" SRC="foot_motif.gif"></A>Introducing a normalisation constant <IMG WIDTH=10 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline1914" SRC="img7.gif">, we can write <IMG WIDTH=429 HEIGHT=34 ALIGN=MIDDLE ALT="tex2html_wrap_inline2400" SRC="img169.gif">. Then again, the binned autocorrelationfunction is defined by<IMG WIDTH=205 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline2402" SRC="img170.gif">. Now wecarry out both integrals and thus eliminate both delta functions. If we choose<IMG WIDTH=10 HEIGHT=7 ALIGN=BOTTOM ALT="tex2html_wrap_inline1914" SRC="img7.gif"> such that <I>C</I>(0)=1, we obtain:<BR><A NAME="eqcspike"> </A><IMG WIDTH=500 HEIGHT=34 ALIGN=BOTTOM ALT="equation1092" SRC="img171.gif"><BR>Thus, all we have to do is to count all possible intervals <IMG WIDTH=41 HEIGHT=21 ALIGN=MIDDLE ALT="tex2html_wrap_inline2408" SRC="img172.gif"> in a bin.The upper panel in Fig. <A HREF="node26.html#figrr">17</A> shows the binned autocorrelation functionwith bin size <IMG WIDTH=61 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline2410" SRC="img173.gif"> sec up to a lag of 6 sec for the heart beat datashown in Fig. <A HREF="node26.html#figrrseries">16</A>. Superimposed is the corresponding curve for asurrogate that has been generated with the deviation from the binnedautocorrelations of the data as a cost function. The two curves arepractically indistinguishable. In this particular case, most of the structureis given by the mean periodicity of the data. The lower trace of the samefigure shows that even a random scramble shows very similar (but not identical)correlations. Information about the main periodicity is already contained inthe distribution of inter-beat intervals which is preserved under permutation.<P>As with unevenly sampled data, the choice of binning and the maximal lag aresomewhat delicate and not that much practical experience exists. It iscertainly again recommendable to avoid empty bins. The possibility to limit thetemporal range of <IMG WIDTH=41 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline2412" SRC="img174.gif"> is a powerful instrument to keepcomputation time within reasonable limits.<P><HR><A NAME="tex2html335" HREF="node27.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif"></A> <A NAME="tex2html333" HREF="node22.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif"></A> <A NAME="tex2html329" HREF="node25.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif"></A> <BR><B> Next:</B> <A NAME="tex2html336" HREF="node27.html">Questions of interpretation</A><B>Up:</B> <A NAME="tex2html334" HREF="node22.html">Various Examples</A><B> Previous:</B> <A NAME="tex2html330" HREF="node25.html">Uneven sampling</A><P><ADDRESS><I>Thomas Schreiber <BR>Mon Aug 30 17:31:48 CEST 1999</I></ADDRESS></BODY></HTML>⌨️ 快捷键说明
复制代码
Ctrl + C
搜索代码
Ctrl + F
全屏模式
F11
切换主题
Ctrl + Shift + D
显示快捷键
?
增大字号
Ctrl + =
减小字号
Ctrl + -