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Increased firing rates monotonically expand neuronal coding bandwidth.

Code ↔ Paper

10 matches between paragraphs of the paper and lines of its authors' code, computed by the harvester (lexical-v1). Click a colored paragraph or line to see its counterpart.

The 10 matches
  1. [1] § Statistical measures, experimental data, and models › Experimental recordings from P-units ↔ profiles/spectral.py, lines 160–283 · score 0.70 · Fourier transformation, stimulus power, spike train, segments, resolution, cross
  2. [2] § Statistical measures, experimental data, and models › Leaky integrate-and-fire model › Analytical result for the coherence function ↔ profiles/spectral.py, lines 103–157 · score 0.63 · cross spectrum, spike train, power spectrum, absolute, susceptibility, coherence
  3. [3] § Statistical measures, experimental data, and models › P-unit model › Fitting process ↔ profiles/baseline.py, lines 107–141 · score 0.63 · serial correlation, baseline activity, interspike intervals
  4. [4] § Statistical measures, experimental data, and models › P-unit model › Fitting process ↔ profiles/punitprofiles.py, lines 316–361 · score 0.61 · steady state firing, step stimuli, onset, intervals, firing rate, baseline
  5. [5] § Statistical measures, experimental data, and models › P-unit model › Simulation parameter ↔ profiles/spectral.py, lines 103–157 · score 0.61 · cross spectra, Spike train, power spectra, resolution, coherence, stimulus
  6. [6] § Statistical measures, experimental data, and models › P-unit model ↔ profiles/punitprofiles.py, lines 442–494 · score 0.60 · generated EOD, EOD frequency, unit model, rectified, amplitude, zero
  7. [7] § Statistical measures, experimental data, and models › Experimental recordings from P-units ↔ profiles/baseline.py, lines 18–56 · score 0.57 · baseline activity, interspike intervals, variation, coefficient, CV
  8. [8] § Statistical measures, experimental data, and models › P-unit model ↔ model.py, lines 43–101 · score 0.54 · amplitude modulations, EOD frequency, thresholding, zero, signal, noise
  9. [9] § Statistical measures, experimental data, and models ↔ profiles/spectral.py, lines 1–18 · score 0.53 · cross spectrum, power spectra, spectral, neuronal, stimulus
  10. [10] § Results › LIF-based P-unit model reproduces experimental observations ↔ profiles/baseline.py, lines 18–56 · score 0.51 · interspike intervals, baseline firing rates, variation, coefficient, CV, spikes

Paper

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The authors' code

Python · 430 lines · 14 KB · GPL-3.0 · 4 matches

  1. """
  2. Spectral analysis of neuronal responses.
  4. ## Functions
  6. - `whitenoise()`: band-limited white noise.
  7. - `rate()`: firing rate computed by kernel convolution.
  8. - `spectra()`: stimulus- and response power spectra, and cross spectrum.
  9. - `susceptibilities()`: stimulus- and response spectra up to second order.
  10. - `diag_projection()`: projection of the chi2 matrix onto its diagonal.
  11. - `hor_projection()`: horizontal projection of the chi2 matrix.
  12. - `peak_size()`: normalized and relative size of a peak expected around a specific frequency.
  14. """
  16. import numpy as np
  17. from scipy.signal import welch, csd
  18. from scipy.stats import norm
  21. def whitenoise(cflow, cfup, dt, duration, rng=np.random.default_rng()):
  22. """Band-limited white noise.
  24. Generates white noise with a flat power spectrum between `cflow` and
  25. `cfup` Hertz, zero mean and unit standard deviation. Note, that in
  26. particular for short segments of the generated noise the mean and
  27. standard deviation of the returned noise can deviate from zero and
  28. one.
  30. Parameters
  31. ----------
  32. cflow: float
  33. Lower cutoff frequency in Hertz.
  34. cfup: float
  35. Upper cutoff frequency in Hertz.
  36. dt: float
  37. Time step of the resulting array in seconds.
  38. duration: float
  39. Total duration of the resulting array in seconds.
  41. Returns
  42. -------
  43. noise: 1-D array
  44. White noise.
  45. """
  46. # number of elements needed for the noise stimulus:
  47. n = int(np.ceil((duration+0.5*dt)/dt))
  48. # next power of two:
  49. nn = int(2**(np.ceil(np.log2(n))))
  50. # indices of frequencies with `cflow` and `cfup`:
  51. inx0 = int(np.round(dt*nn*cflow))
  52. inx1 = int(np.round(dt*nn*cfup))
  53. if inx0 < 0:
  54. inx0 = 0
  55. if inx1 >= nn/2:
  56. inx1 = nn/2
  57. # draw random numbers in Fourier domain:
  58. whitef = np.zeros((nn//2+1), dtype=complex)
  59. # zero and nyquist frequency must be real:
  60. if inx0 == 0:
  61. whitef[0] = 0
  62. inx0 = 1
  63. if inx1 >= nn//2:
  64. whitef[nn//2] = 1
  65. inx1 = nn//2-1
  66. phases = 2*np.pi*rng.random(size=inx1 - inx0 + 1)
  67. whitef[inx0:inx1+1] = np.cos(phases) + 1j*np.sin(phases)
  68. # inverse FFT:
  69. noise = np.real(np.fft.irfft(whitef))
  70. # scaling factor to ensure standard deviation of one:
  71. sigma = nn / np.sqrt(2*float(inx1 - inx0))
  72. return noise[:n]*sigma
  75. def rate(time, spikes, sigma):
  76. """ Firing rate computed by kernel convolution.
  78. Parameters
  79. ----------
  80. time: ndarray of float
  81. Times at which firing rate is evaluated.
  82. spikes: list of ndarray of float
  83. Spike times.
  84. sigma: float
  85. Width of the Gaussian kernel as a standard deviation.
  87. Returns
  88. -------
  89. rate: ndarray of float
  90. Firing rate of convolved spike trains averaged over trials.
  91. ratesd: ndarray of float
  92. Corresponding standard deviation.
  93. """
  94. kernel = norm.pdf(time[time < 8*sigma], loc=4*sigma, scale=sigma)
  95. rates = np.zeros((len(spikes), len(time)))
  96. xtime = np.append(time, time[-1] + time[1] - time[0])
  97. for i, spiket in enumerate(spikes):
  98. b, _ = np.histogram(spiket, xtime)
  99. rates[i] = np.convolve(b, kernel, 'same')
  100. return np.mean(rates, 0), np.std(rates, 0)
  103. def spectra(stimulus, spikes, dt, nfft):
  104. """Stimulus- and response power spectra, and cross spectrum.
  106. Compute the complex-valued transfer function (first-order
  107. susceptibility) and the stimulus-response coherence like this:
  109. ```
  110. freqs, pss, prr, prs = spectra(stimulus, spikes, dt, nfft)
  111. transfer = prs/pss
  112. coherence = np.abs(prs)**2/pss/prr
  113. ```
  115. The gain of the transfer function is the absolute value of the
  116. transfer function:
  118. ```
  119. gain = np.abs(prs)/pss
  120. ```
  122. Parameters
  123. ----------
  124. stimulus: ndarray of float
  125. Stimulus waveform with sampling interval 'dt'.
  126. spikes: list of ndarrays of float
  127. Spike times in response to the stimulus.
  128. dt: float
  129. Sampling interval of stimulus and resolution of the binary spike train.
  130. nfft: int
  131. Number of samples used for each Fourier transformation.
  133. Returns
  134. -------
  135. freqs: ndarray of float
  136. The frequencies corresponding to the spectra.
  137. pss: ndarray of float
  138. Power spectrum of the stimulus.
  139. prr: ndarray of float
  140. Power spectrum of the response averaged over trials.
  141. prs: ndarray of complex
  142. Cross spectrum between stimulus and response averaged over trials.
  144. """
  145. time = np.arange(len(stimulus))*dt
  146. freq, pss = welch(stimulus, fs=1/dt, nperseg=nfft, noverlap=nfft//2)
  147. prr = np.zeros((len(spikes), len(freq)))
  148. prs = np.zeros((len(spikes), len(freq)), dtype=complex)
  149. for i, spiket in enumerate(spikes):
  150. b, _ = np.histogram(spiket, time)
  151. b = b / dt
  152. f, rr = welch(b - np.mean(b), fs=1/dt, nperseg=nfft, noverlap=nfft//2)
  153. f, rs = csd(b - np.mean(b), stimulus,
  154. fs=1/dt, nperseg=nfft, noverlap=nfft//2)
  155. prr[i] = rr
  156. prs[i] = rs
  157. return freq, pss, np.mean(prr, 0), np.mean(prs, 0)
  160. def susceptibilities(stimulus, spikes, dt=0.0005, nfft=2**9, nmax=0):
  161. """Stimulus- and response spectra up to second order.
  163. Compute the complex-valued transfer function (first-order
  164. susceptibility) and the stimulus-response coherence like this:
  166. ```
  167. freqs, pss, prr, prs, prss, n = susceptibilities(stimulus, spikes, dt, nfft)
  168. transfer = prs/pss
  169. coherence = np.abs(prs)**2/pss/prr
  170. ```
  172. The gain of the transfer function is the absolute value of the
  173. transfer function and has the unit Hz/[s]:
  175. ```
  176. gain = np.abs(prs)/pss
  177. ```
  179. The complex-valued second-order susceptibility has the unit
  180. Hz/[ss] and can be computed like this:
  182. ```
  183. chi2 = prss*0.5/(pss.reshape(1, -1)*pss.reshape(-1, 1))
  184. ```
  186. The variance of the stimulus is the integral over the stimulus
  187. power spectral density `pss` (unit [s]^2/Hz):
  188. ```
  189. deltaf = freqs[1] - freqs[0] # same as 1/(dt*nfft)
  190. vars = np.sum(pss)*deltaf
  191. ```
  192. Likewise for the response.
  194. The response spectral density `prr` (in Hz^2/Hz = Hz) approaches
  195. the firing rate for large frequencies.
  197. Parameters
  198. ----------
  199. stimulus: ndarray of float
  200. Stimulus waveform with sampling interval 'dt'.
  201. spikes: list of ndarrays of float
  202. Spike times in response to the stimulus.
  203. dt: float
  204. Sampling interval of stimulus and resolution of the binary spike train.
  205. nfft: int
  206. Number of samples used for each Fourier transformation.
  207. nmax: int
  208. Maximum number of FFT segments to be used. If 0, use all segments.
  210. Returns
  211. -------
  212. freqs: ndarray of float
  213. The frequencies corresponding to the spectra.
  214. pss: ndarray of float
  215. Power spectral density of the stimulus in unit [s]^2/Hz.
  216. prr: ndarray of float
  217. Power spectral density of the response averaged over segments
  218. in unit Hz^2/Hz = Hz.
  219. prs: ndarray of complex
  220. Cross spectrum between stimulus and response averaged over segments
  221. in unit Hz[s]/Hz = [s].
  222. prss: ndarray of complex
  223. Cross bispectrum between stimulus and response averaged
  224. over segments in unit [s]^2/Hz.
  225. n: int
  226. Number of FFT segments used.
  228. """
  229. freqs = np.fft.fftfreq(nfft, dt)
  230. freqs = np.fft.fftshift(freqs)
  231. f0 = np.argmin(np.abs(freqs)) # index of zero frequency
  232. fidx = np.arange(len(freqs))
  233. fsum_idx = fidx.reshape(-1, 1) + fidx.reshape(1, -1) - f0
  234. fsum_idx[fsum_idx < 0] = 0
  235. fsum_idx[fsum_idx >= len(fidx)] = len(fidx) - 1
  236. f0 = len(freqs)//4
  237. f1 = 3*len(freqs)//4
  238. segments = range(0, len(stimulus) - nfft, nfft)
  239. # stimulus:
  240. p_ss = np.zeros(len(freqs))
  241. fourier_s = np.zeros((len(segments), len(freqs)), complex)
  242. n = 0
  243. for j, k in enumerate(segments):
  244. fourier_s[j] = np.fft.fft(stimulus[k:k + nfft], n=nfft)
  245. fourier_s[j] = np.fft.fftshift(fourier_s[j])
  246. p_ss += np.abs(fourier_s[j]*np.conj(fourier_s[j]))
  247. n += 1
  248. if nmax > 0 and n >= nmax:
  249. break
  250. scale = dt/nfft/n
  251. p_ss *= scale
  252. # response spectra:
  253. time = np.arange(len(stimulus))*dt
  254. p_rr = np.zeros(len(freqs))
  255. p_rs = np.zeros(len(freqs), complex)
  256. p_rss = np.zeros((len(freqs), len(freqs)), complex)
  257. n = 0
  258. for i, spiket in enumerate(spikes):
  259. b, _ = np.histogram(spiket, time)
  260. b = b / dt
  261. for j, k in enumerate(segments):
  262. # stimulus:
  263. fourier_s1 = fourier_s[j].reshape(len(fourier_s[j]), 1)
  264. fourier_s2 = fourier_s[j].reshape(1, len(fourier_s[j]))
  265. fourier_s12 = np.conj(fourier_s1)*np.conj(fourier_s2)
  266. # response:
  267. fourier_r = np.fft.fft(b[k:k + nfft] - np.mean(b), n=nfft)
  268. fourier_r = np.fft.fftshift(fourier_r)
  269. p_rr += np.abs(fourier_r*np.conj(fourier_r))
  270. p_rs += np.conj(fourier_s[j])*fourier_r
  271. p_rss += fourier_s12*fourier_r[fsum_idx]
  272. n += 1
  273. if nmax > 0 and n >= nmax:
  274. break
  275. if nmax > 0 and n >= nmax:
  276. break
  277. scale = dt/nfft/n
  278. freqs = freqs[f0:f1]
  279. p_ss = p_ss[f0:f1]
  280. p_rr = p_rr[f0:f1]*scale
  281. p_rs = p_rs[f0:f1]*scale
  282. p_rss = p_rss[f0:f1, f0:f1]*dt*scale
  283. return freqs, p_ss, p_rr, p_rs, p_rss, n
  286. def diag_projection(freqs, chi2, fmax):
  287. """ Projection of the chi2 matrix onto its diagonal.
  289. Adapted from https://stackoverflow.com/questions/71362928/average-values-over-all-offset-diagonals
  291. Parameters
  292. ----------
  293. freqs: ndarray of float
  294. Frequencies of the chi2 matrix.
  295. chi2: 2-D ndarray of float
  296. Second-order susceptibility matrix.
  297. fmax: float
  298. Maximum frequency for the projection.
  300. Returns
  301. -------
  302. dfreqs: ndarray of float
  303. Frequencies of the projection.
  304. diagp: ndarray of float
  305. Projections of the chi2 matrix onto its diagonal.
  306. That is, averages over the anti-diagonals.
  307. """
  308. i0 = np.argmin(freqs < 0)
  309. i1 = np.argmax(freqs > fmax)
  310. if i1 == 0:
  311. i1 = len(freqs)
  312. chi2 = chi2[i0:i1, i0:i1]
  313. n = chi2.shape[0]
  314. diagp = np.zeros(n*2-1, dtype=float)
  315. for i in range(n):
  316. diagp[i:i + n] += chi2[i]
  317. diagp[0:n] /= np.arange(1, n+1, 1, dtype=float)
  318. diagp[n:] /= np.arange(n-1, 0, -1, dtype=float)
  319. dfreqs = np.arange(len(diagp))*(freqs[i0 + 1] - freqs[i0]) + freqs[i0]
  320. return dfreqs, diagp
  323. def hor_projection(freqs, chi2, fmax):
  324. """ Horizontal projection of the chi2 matrix.
  326. Parameters
  327. ----------
  328. freqs: ndarray of float
  329. Frequencies of the chi2 matrix.
  330. chi2: 2-D ndarray of float
  331. Second-order susceptibility matrix.
  332. fmax: float
  333. Maximum frequency for the projection.
  335. Returns
  336. -------
  337. hfreqs: ndarray of float
  338. Frequencies of the projection.
  339. horp: ndarray of float
  340. Projections of the chi2 matrix onto its x-axis.
  341. That is, averages over columns.
  342. """
  343. i0 = np.argmin(freqs < 0)
  344. i1 = np.argmax(freqs > fmax)
  345. if i1 == 0:
  346. i1 = len(freqs)
  347. hfreqs = freqs[i0:i1]
  348. chi2 = chi2[i0:i1, i0:i1]
  349. horp = np.mean(chi2, 1)
  350. return hfreqs, horp
  353. def peak_size(freqs, spectrum, ftarget, median=True,
  354. searchwin=50, distance=10, averagewin=10):
  355. """Normalized and relative size of a peak expected around a specific frequency.
  357. The peak is searched within `searchwin` Hz around the target
  358. frequency. As a baseline amplitude of the spectrum either the
  359. averaged values of the spectrum within `averagewin` Hz at a
  360. distance of `distance` Hz to the left and right of the found
  361. peak frequency (`median` is `False`, default), or the median of
  362. the whole spectrum is taken (`median` is `True`).
  364. Parameters
  365. ----------
  366. freqs: ndarray of float
  367. Frequencies of the spectrum.
  368. spectrum: ndarray of float
  369. Some spectrum. Or a projection of the chi2 matrix.
  370. ftarget: float
  371. The frequency where a peak is expected in the spectrum.
  372. median: bool
  373. If True, normalize the peak height by the median of the spectrum.
  374. Otherwise (default), normalize by averaged values of the spectrum
  375. close to the `peak frequency.
  376. searchwin: float
  377. Search for the largest peak in the spectrum at `ftarget` plus and
  378. minus `searchwin` Hertz.
  379. distance: float
  380. The windows for estimating the baseline around the peak start
  381. `distance` Hertz to the left and right of the found peak.
  382. averagewin: float
  383. For estimating the baseline around the peak, an average is taken in two
  384. `averagewin` Hertz wide windows that are located `distance`
  385. Hertz to the left and right of the detected peak.
  387. Returns
  388. -------
  389. peak_norm: float
  390. The height of the peak close to `ftarget` normalized to the
  391. baseline around the peak.
  392. peak_rel: float
  393. The height of the peak close to `ftarget` relative to the
  394. baseline around the peak.
  395. peak_freq: float
  396. The frequency of the detected peak.
  398. """
  399. mask = (freqs > ftarget - searchwin) & (freqs < ftarget + searchwin)
  400. snippet = spectrum[mask]
  401. if len(snippet) == 0:
  402. return np.nan, np.nan, np.nan
  403. peak = np.max(snippet)
  404. fpeak = freqs[np.argmax(snippet) + np.argmax(mask)]
  405. bleft = np.nan
  406. bright = np.nan
  407. baseline = np.nan
  408. if median:
  409. baseline = np.median(spectrum)
  410. else:
  411. mask = (freqs >= fpeak - distance - averagewin) & \
  412. (freqs <= fpeak - distance)
  413. bleft = np.mean(spectrum[mask]) if np.sum(mask) > 0 else np.nan
  414. mask = (freqs >= fpeak + distance) & \
  415. (freqs <= fpeak + distance + averagewin)
  416. bright = np.mean(spectrum[mask]) if np.sum(mask) > 0 else np.nan
  417. if np.isfinite(bleft) and np.isfinite(bright):
  418. baseline = 0.5*(bleft + bright)
  419. elif np.isfinite(bleft):
  420. baseline = bleft
  421. elif np.isfinite(bright):
  422. baseline = bright
  423. else:
  424. baseline = np.nan
  425. if np.isnan(baseline):
  426. return np.nan, np.nan, np.nan
  427. peak_norm = peak/baseline
  428. peak_rel = peak - baseline
  429. return peak_norm, peak_rel, fpeak

spectral.py at commit 08b9dbd, under GPL-3.0 · at the source

Overview

Authors: Alexander Wendt1, Kolja Klett2,3, Jan Benda1,4, Benjamin Lindner2,3, Jan Grewe1
ORCID iDs: Alexander Wendt, Kolja Klett, Jan Benda, Benjamin Lindner, Jan Grewe
  1. Neuroethology, University of Tübingen,Auf der Morgenstelle 28, 72076 Tübingen, Germany
  2. Physics Department, Humboldt University Berlin,Newtonstr. 15, 12489 Berlin, Germany
  3. Bernstein Center for Computational Neuroscience Berlin,Philippstr. 13, Haus 2, 10115 Berlin, Germany
  4. Bernstein Center for Computational Neuroscience Tübingen,Maria-von-Linden-Straße 6, 72076 Tübingen, Germany
Institutions: University of Tübingen (Germany); Humboldt-Universität zu Berlin (Germany); Bernstein Center for Computational Neuroscience Berlin (Germany); Bernstein Center for Computational Neuroscience Tübingen (Germany)
Journal: Biological cybernetics, volume 120, issue 5-6, article 35
Dates: received 14 April 2026; accepted 27 August 2026; published online 17 September 2026; in print 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.1007/s00422-026-01063-3 · PMID 42749854 · PMCID PMC13582337 · OpenAlex W7213424585
Open access: hybrid, a free copy (OpenAlex)
Status: code verified
Categories: none (in silico) (organism), computational (subfield)
Methods: Spectral & time-frequency, Connectivity, Single-unit activity, calcium imaging
Keywords: Spiking neurons, LIF model, Encoding, Bandwidth, Coherence
MeSH: Action Potentials*, Models, Neurological*, Neurons*, Animals, Computer Simulation (* major topic)
Topic: Fish biology, ecology, and behavior (Nature and Landscape Conservation, Environmental Science), according to OpenAlex
Funding: Deutsche Forschungsgemeinschaft (430157666)
Citations: not cited yet (Europe PMC); 67 references in the paper

Abstract

Sensory information is often processed by populations of neurons that vary with respect to their levels of spiking activity. It appears plausible that neurons that fire more often are capable of following faster changes in the input signal than those with lower firing rates. In this study, we test this intuitive assumption by systematically investigating how the spiking activity of a neuron affects information transmission. To this end, we employ the coherence function as a spectral measure of information transmission and study the relation between the width of the coherence function, which we call coding bandwidth, and the firing rate. The reexamination of experimental data from a previous study on weakly-electric fish and the simulation of biologically inspired neuron models fitted to real neurons reveal indeed a significant correlation between firing rate and coding bandwidth. In addition, known analytical expressions for the coherence function for the stochastic leaky integrate-and-fire model demonstrate a linear relation for high firing rates. However, in contrast to intuition, the width of the coherence function becomes independent of the firing rate and is completely set by the membrane time constant for very low firing rates.

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doi.gin.g-node.org/10.12751/g-node.5b08du

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Wendt, A., Klett, K., Benda, J., Lindner, B., & Grewe, J. (2026). Increased firing rates monotonically expand neuronal coding bandwidth. Biological cybernetics, 120(5-6), 35. https://doi.org/10.1007/s00422-026-01063-3

BibTeX

@article{wendt2026increased,
author = {Wendt, Alexander and Klett, Kolja and Benda, Jan and Lindner, Benjamin and Grewe, Jan},
title = {{Increased firing rates monotonically expand neuronal coding bandwidth}},
journal = {Biological cybernetics},
year = {2026},
month = sep,
volume = {120},
number = {5-6},
pages = {35},
publisher = {Springer Science+Business Media},
issn = {0340-1200},
doi = {10.1007/s00422-026-01063-3},
url = {https://doi.org/10.1007/s00422-026-01063-3},
pmid = {42749854},
pmcid = {PMC13582337}
}

RIS

TY - JOUR
AU - Wendt, Alexander
AU - Klett, Kolja
AU - Benda, Jan
AU - Lindner, Benjamin
AU - Grewe, Jan
TI - Increased firing rates monotonically expand neuronal coding bandwidth
T2 - Biological cybernetics
J2 - Biol Cybern
PY - 2026
DA - 2026/09/17
VL - 120
IS - 5-6
SP - 35
SN - 0340-1200
PB - Springer Science+Business Media
DO - 10.1007/s00422-026-01063-3
UR - https://doi.org/10.1007/s00422-026-01063-3
LA - en
ER -

CSL-JSON

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"container-title-short": "Biol Cybern",
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