Fig 1.
Spread of conduction delays depends on population size.
A: Action potential traveling takes time. Depending on the conduction velocity of the nerve fibre (vcond.) action potential arrivals at the target will be delayed depending on the travel distance. Bi—Di: Sketches of the integration of inputs from three differently sized receptive fields (blue ellipses). All neurons of the presynaptic population are driven by the same stimulus (s(t)) and project to an integrating neuron. The length of the axons connecting the pre- and postsynaptic neurons will vary with the receptor’s location in the receptive field. Bii—Dii: Depending on the spatial extent of the receptive field, which is proportional to the size of the presynaptic population, the information from one end of the population will reach the integrating neuron earlier than from the other end. Viewed from the center of the population, the first will lead while the latter will lag behind (green and red areas, respectively). Marginal plots show the probability density of the respective delays among the postsyanptic neuron’s inputs.
Fig 2.
Analyzes on baseline and stimulus driven responses.
A—C: Baseline response of an example P-unit. A: Short segments of the fish’s self-generated EOD (top) and the simultaneously recorded neuronal activity of a P-unit (bottom) in the absence of an external stimulus. This example neuron had a spontaneous activity of 189 Hz. B: Spike times are precisely timed within the EOD period. The average EOD waveform is depicted in orange and the blue histogram shows the phases in which action potentials occur. From this we calculated the vector strength (VS, Eq 1) which is 0.83 for this neuron. From the circular mean of the locking histogram we calculated the delay between EOD onset (time of the rising zero-crossing) and spiking activity. C: P-units skip EOD-cycles and sometimes fire bursts of action potentials. The interspike interval histogram shows the typical wide and multimodal distribution. Response regularity is characterized by the coefficient of variation (Eq 2, 0.61 in this example). The burstiness indicates the proportion of spikes occurring in intervals less than 1.5 times the EOD period [51] (0.11 in this example). D:—F: Driven responses of the same P-unit. D: The responses to frozen sequences of band-limited white noise of EOD amplitude modulations (0—300 Hz, top trace) are used to characterize the stimulus encoding properties. The stimulus profile is encoded in the spiking activity of the neuron (raster plot for 5 consecutive trials and firing rate in blue, bottom). The firing rate modulates around the baseline firing rate and the cell’s sensitivity is characterized by the strength of the modulation (Eq 4). The shaded area around the mean depicts the response variability, i.e. the across trial standard deviation of the firing rate. E: From the transfer function (Eq 6) we estimate the maximum gain as well as the upper and lower cutoff frequencies (red dashed vertical lines). F: Similar measures were extracted from the stimulus response coherence (Eq 7). The integral of the coherence spectrum is used a lower bound estimation of the mutual information between stimulus and response (Eq 8). The spectra in E and F have been smoothed with an 11 point running average. Abbreviations: VS, vector strength; CV, coefficient of variation; ISI, interspike interval; coh, coherence.
Table 1.
Parameters in the membrane model part refer to Eq 9.
Fig 3.
Correlation of baseline response properties and receptor position.
r-values are the Pearson correlation coefficients, p-values are Bonferroni corrected. Dashed thin line shows the mean and gray areas the standard deviation estimated in bins of 10% width. A: Baseline firing rates estimated with only the unperturbed fish’s field present. Inset: In this study 84 p-type electroreceptor afferents were recorded. The blue histogram in the inset shows the distribution of receptor positions along the rostro-caudal body axis expressed relative to the total body length. The cumulative (red line) indicates that three fourths of the recorded cells were recorded in the frontal third of the body. Due to the recording location cells in the frontal 20% of the body could not be recorded. B: Coefficient of variation of the interspike interval distribution (CVISI) during baseline activity. C: Fraction of action potentials observed as part of action potential bursts as previously defined [51]. D: Vector strength quantifying the phase-locking to the EOD.
Fig 4.
Correlation of stimulus driven response properties and receptor position.
r-values are the Pearson correlation coefficients, p-values are Bonferroni corrected. Dashed thin line shows the mean and gray areas the standard deviation estimated in bins of 10% width.A: Response modulation of the across-trial average firing rate (Eq 4) in response to frozen white noise amplitude modulations. Please note that these stimuli were only presented to a subset of 39 neurons which could be recorded sufficiently long to allow for position estimation and stimulation with white noise stimuli and were stimulated with the same stimulus intensity (10% contrast). B: response variability, i.e. the across-trial standard deviation of the firing rate (Eq 5). C, D: lower and upper -3 dB cutoff frequencies of the transfer function. E: peak gain of transfer function. F: Mutual information between responses and the stimulus estimated from the stimulus response coherence spectrum.
Fig 5.
Preferred phase and response delay depend on receptor position.
A: Top: Phase relation between the baseline spikes and the fish’s own EOD measured at the operculum (Reference EOD, S1 Fig). Colors represent clusters as established by K-means clustering. Crosses depict cluster centroids. Middle: Correlation of preferred phase and receptor position after the caudal (blue) cluster was shifted by one EOD cycle (2π). Now phases correlate positively with the rostro-caudal receptor position (solid line, Pearson correlation). Bottom: Re-plotting of the data in absolute units. Spike phases are expressed as the absolute delay between the beginning of the EOD (ascending zero crossing) and the respective action potential. The receptor positions on the rostro-caudal axis are given in absolute numbers relative to the snout position. The slope of the regression line is the inverse of the conduction velocity of 47.2. B: Delay between the cellular response and the white noise stimulus estimated from the peak of spike triggered average stimulus (STA) for those cells in which white noise responses could be recorded (n = 39). C: Correlation of the STA delay and the absolute phase delay.
Fig 6.
Information content of homogeneous and heterogeneous populations of P-units.
A: Mutual information estimated from the stimulus response coherence (Eq 8) carried by populations of increasing size. Blue lines show information content of homogeneous populations created by combining randomly chosen trials of the same neuron. Since not all recordings have the same number of stimulus repetitions, the maximum possible population size varies between recorded neurons. Thick blue line is the average across all homogeneous populations. Red dots show the average information estimated from heterogeneous populations (see Methods for details). Error bars indicate the standard deviation, thin red lines depict the best and worst performance observed in the sample of heterogeneous populations. Tests on statistically significant differences in mutual information in homogeneous and heterogeneous population were performed using a Mann Whitney U-Test with Bonferroni correction for the highlighted population sizes, one asterisk: α < 5% two stars indicate α < 1%. B: Comparison of the mutual information carried by homogeneous (x-axis) and heterogeneous (y-axis) populations that are similarly driven by the stimulus (response modulation of heterogeneous population response within a ±20% range of the respective homogeneous response) for three different population sizes. Large symbols depict the center of gravity of the respective distributions. Dashed line is the identity line. Abbreviations: m.i., mutual information; het., heterogeneous; hom., homogeneous.
Fig 7.
Conduction delays in a population act as an information filter.
A: Average mutual information carried by heterogeneous populations of increasing population size that are subjected to varying simulated conduction delays. B: Mutual information as a function of conduction delay compared for the three different spectral bands. High frequency information is more strongly affected than low-frequency content. Mutual information is normalized to the zero-delay performance, a population size of 20 neurons was chosen here.
Fig 8.
Conduction velocity and stimulus dynamics constrain meaningful population sizes of LIF model neurons.
A—C Mutual information between stimulus and response estimated from the population responses of LIF-model neurons (see sketches in Fig 1A). These models assume 1D populations with a density of 2000 neurons per m of the sensory surface. Model parameterization was completely arbitrary and identical for all model neurons (see Methods). The models were driven with the same frozen noise sequences with spectral power in the ranges 0–100 Hz, 100–200 Hz and 200–300 Hz (dashed, dash-dotted, and dotted lines, respectively). Each model had the same amount of (independent) noise added to the driving stimulus. Three realistic conduction velocities were simulated and compared to the unrealistic instantaneous conduction (blue solid line in all figures, “without delay”). With decreasing conduction velocity the encoding performance drops at smaller populations. High-frequency encoding is most sensitive to spread in conduction delays. The “harmonic” structure seen in A is a consequence of the stimulus’ auto-correlation. A: Conduction velocity of 7 ms−1, as described for some visual fibers in the monkey corpus callosum [55]B: 25 ms−1 for example measured in squid giant axons [18], C: 50 ms−1 as estimated above for P-unit afferents.
Fig 9.
Information filtering by conduction delays is stronger than by synaptic filtering.
Comparison of the effects of conduction delays (x-axis) and synaptic transfer (y-axis) on the information content of the population response (color code). Dashed black line is the identity line. Area between horizontal dashed lines depicts the standard deviations (σkernel) of the fast and slow EPSP components as reported for the P-unit to pyramidal cell synapses [70]. Vertical dashed lines enclose the range of conduction delays that can be expected for pyramidal neurons in the lateral segment of the electrosensory lateral line lobe [39] considering the 2σ or 4σ receptive field sizes. The data shown here assumes a population size of 16 cells.