Fig 1.
The spike threshold depends on the history of the membrane potential in both real and simulated data.
(A) We performed patch-clamp recordings in layer 2/3 pyramidal neurons in vitro, in response to population input from stimulation in layer 4 (left). The pyramidal neuron identity was confirmed in a subset by filling the targeted neuron using biotin (middle). (B) From recorded action potentials (top), the spike threshold is determined as the maximal positive peak of the second derivative of the membrane potential (bottom). (C1) A cortical neuron stimulated with current inputs of different slopes (bottom, different shades of gray) lead to action potentials (top, corresponding grays) with different thresholds for spike initiation (top, red lines in corresponding brightness to grays of voltage traces, inset shows zoom in of spike initiation). The response is delayed w.r.t. to the stimulation due to the propagation delay from L4 to L2/3. The inset shows a magnified view of the threshold region. (C2) As in previous studies, thresholds were found to vary with the slope of the preceding membrane voltage. In the current stimulation settings, only a limited range of input slopes was realized. (D1) Neurons with an adaptive threshold were simulated on the basis of the model by Fontaine et al. [2], after adapting the parameterization to cortical excitatory neurons (see Methods). In addition to the voltage traces (grays), the adapting thresholds are also shown (reds, brightness corresponding to the gray traces). (D2) Applying the same analysis as in the in vitro data to measure the threshold, indicates that designed and measured threshold agree. The relationship between EPSP slope and spike threshold is overall captured by an exponential function especially when the wider range of EPSP slopes was used, which could be explored in the model (compare C2 and D2), see also [5]. (E1) Neurons with a fixed threshold were also simulated. The threshold was set to equalize firing probability with the adaptive threshold model. (E2) Re-estimating the threshold, we obtain the expected constant threshold.
Fig 2.
The input-output relationship sharpens due to the adaptive threshold.
In previous studies, the dependence of spike threshold on previous membrane potential dynamics was most frequently linked to EPSP amplitude (AEPSP) or slope (SlEPSP). Both measures can be reliably predicted by the combination of temporal input spread (σ) and the number of contributing neurons (Ninputs) in point neurons. (A) The AEPSP (data: gray, mean: red) is predicted with small standard deviation (S.D.) (maroon) by σ and Ninputs across a wide range of values for the adaptive threshold model (A1). Prediction quality was similarly good for the fixed threshold model (mean: blue, S.D.: dark blue, A2). (B) Similarly, the SlEPSP amplitude is predicted well with small S.D. on the basis of σ and Ninputs, for both adaptive (B1) and fixed (B2). (C) Spike probability follows generally a similar shape as a function of σ and Ninputs. However, the speed of transition between spiking and non-spiking domain is overall greater for the adaptive threshold model (C1 vs.C2), translating into a steeper decision criterion as a function of the input parameters. (D) Spike probability as a function of only σ, is well fitted by a sigmoid, with the adaptive model (D1 vs. D2, for Ninput = 37) exhibiting a steeper slope as a function of different σ’s (adaptive: s = 0.18, fixed: s = 0.28) as well as a lower midpoint, indicating overall operation on a faster time-scale However, the inverse is the case for the dependence on Ninputs (E1 vs. E2, for σ = 2.5 ms). Error bars represent 2x SEMs and are barely visible, only 1/50 of points plotted in A/B to improve display.
Fig 3.
The adaptive threshold neuron is more informative for high temporal precision and low noise than the fixed threshold neuron.
Two types of encoding (left) were investigated on the input side, either a classical rate encoding (top), where the number of input spikes carry the information but spike times are drawn randomly from a normal distribution centered at t0, and a pattern encoding (bottom), where the spatiotemporal pattern of inputs encodes the information and spike times are hence precise across repetitions. In both cases, the temporal precision (σ) according to which the stimuli are drawn is important (bottom). (A) Responses of the adaptive threshold model neurons (red) encode information mostly at low temporal spread σ, while the fixed threshold neurons (blue) possess a wider range of encoding w.r.t. to σ if information is rate-encoded. (B) This relationship holds across a wide range of noise inputs, with adaptive threshold neurons encoding generally better for lower values of σ, and fixed threshold neurons for higher (B3, the differences in mutual information between adaptive and fixed threshold models). Color mapping here represents information. Noise was modeled as independent Poisson spike trains with constant rate for each input neuron; there were 100 input neurons in total. (C) A similar relationship in information encoding between the models is observed when information is decoded from the temporal pattern of incoming spikes. Again the adaptive threshold model performs better for low σ. (D) This relationship holds only for a limited range of noise, after which point the differences between the models becomes fairly small.
Fig 4.
Membrane and threshold time constants influence temporal selectivity similarly.
(A-B) For rate encoding, the firing rate of the model neurons was limited by τm (A) and τθ (B), if either of them were too small. Short τm prevent integration, while short τθ lead to a threshold that quickly follows the input. (C-D) The limitation in firing rate by τm (C) and τθ (D) translates to a limitation of represented stimulus information. Only for short values of τm and τθ the lowpass shape gives way for a bandpass shape, which is a consequence of the decoding bin size chosen here (2ms): for small σ and low τm/τθ the responses to different stimuli will tend to fall in a small number of bins. (E) We extract the average σ that maximized MI for a combination of τm and τθ. The resulting dependence is monotonically in τm and τθ, consistent with a change in the σ edge of the lowpass relationship (Fig 3A). (F-G) For pattern encoding, the firing rate is limited analogously by τm (F) and τθ (G). (H-I) The value of σ that leads to highest MI shifts with both τm (H) and τθ (I), indicating a match between the membrane dynamics and the temporal scale of input patterns that can effectively be encoded. Larger σ values generate temporal patterns that are more distinguishable but the spikes are more dispersed in time,and both τm and τθ set the temporal limit in which the spikes can be integrated. (J) The best MI is achieved at different average σ across the range of τm and τθ, monotonically increasing for both parameters.
Fig 5.
Information decoding across different membrane states and spike threshold types for a ‘stimulus-centered’ time reference.
(A) For small differences in state (A1: -62mV vs. -65mV) both adaptive (red) and fixed (blue) threshold models show a shallow dependence on stimulus strength (different rates, range [50:2:60] inputs, total stimulus entropy ~ 2.58 bit, dark color = 60, light color = 50 EPSPs). The adaptive threshold model compensates partly for the difference in initial voltage, and thus exhibits smaller differences in spiking behavior across states. During larger fluctuations in membrane potential (A2: -62mV vs. -72mV) this behavior is qualitatively retained. The different states also scale the slope (average spike times vs. input strength) and spike time variability for each model, with the fixed model being influenced more strongly in both cases. NS, non-spiking trials. (B) Mutual information is estimated across all states with the state known (light colors) or unknown (dark colors) to the decoder. For small state differences (B1) the adaptive model’s MI is independent of state knowledge (RI close to 1, bottom), while the fixed model shows a strong dependence to membrane state (RI around 0.6, indicating ~40% of MI added by state knowledge). For larger state differences, this relationship inverts (B2). Note that in B2 upper panel the dark blue curve almost fully overlaps with the light blue curve. (C) To understand this inversion, we relate the response distributions to state difference. The correlation coefficient between response PSTHs measures similarity across state differences. The adaptive threshold neuron (red) retains similar responses for larger state differences than the fixed threshold neuron (blue). (D) The correlation coefficient of response distributions predicts qualitatively the RI values, which indicates the advantage of knowing state during decoding. (E) The adaptive model encodes information in a state-independent manner for small state differences, while the fixed threshold model becomes more state-independent only for larger state differences. This switch is caused by a shift from overlapping to non-overlapping temporal decoding ranges, as indicated by the correlation coefficients of the PSTHs across different states (see Fig 5C, where a vanishing correlation indicates a lack of overlap between the responses across states). Hence, the adaptive threshold model compensates for a part of the initial state, however, does not encode more information independently if a stimulus-based, fixed-time decoding reference is used. Error bars indicate SEMs across all state differences of a given size.
Fig 6.
An adaptive threshold neuron represents information more robustly across membrane states for a ‘response-centered’ time reference.
Since stimulus onset is not known internally, a population-based decoding reference has been suggested to serve a biologically relevant surrogate for stimulus timing [25]. With this reference, spike-timing is measured relative to the peak-time of the population response, i.e. local maxima of the population peristimulus time histogram (PSTH). (A1) For small differences in state, adaptive and fixed threshold models show little difference in their relative-time response and consequently the contribution of the knowledge about the state becomes negligible (B1). All colors are as in the preceding figure. (A2) For larger state differences, the response time distributions now differ significantly in their variance, with the fixed threshold model exhibiting a much larger increase in spread for the more hyperpolarized membrane state. Consequently, decoding across states becomes less robust for the fixed model than for the adaptive threshold model (B2). (C) While the ‘response-centered’ time reference increases the similarity of the PSTH across states for both the fixed and the adaptive threshold model, the latter profits more, widening the gap between the models for larger state differences. (D) As before, the correlation coefficient between the PSTHs remains a good predictor for the RI values. (E) In the case of the moving decoding reference, the adaptive model is generally more robust than the fixed threshold model across all state differences investigated, as indicated by a higher RI value. Error bars indicate SEMs across all state differences of a given size.
Fig 7.
The state dependence of neural data corresponds closely to adaptive threshold behavior.
Identical analyses were carried out as for the model data in the preceding figures. (A) Across three initial states of voltage (-80, -70, -60mV) a correlation between threshold and the EPSP slope was observed, i.e. a negative dependence between spike threshold and EPSP slope. Red lines, least-square linear fit to the data. (B) All recorded neurons (N = 11) exhibited this behavior. The average slopes across the different states were across the three states, i.e. -0.99 (0.27), -0.92 (0.19), and 0.90 (0.23) ms, respectively. Numbers in () are s.d. Red lines, average across all the neurons. (C) For small state differences, both the response patterns (C1), the decoded information (D1 top) and the robustness (D1 bottom, measured as robustness index RI) remain comparable across stimulus-centered (orange) and response-centered (red) decoding. N.S., non-spiking trials. (D) For larger state differences the advantage of decoding with an adaptive threshold becomes evident (D2). Stim. cent., stimulus-centered; Resp. cent., response-centered. (E) The similarity of PSTHs as a function of state difference reflects the behavior of the adaptive threshold model (Figs 5C and 6C, red) exhibiting a slow decay, based on a similar robustness in mean and variance of the spike-timing. (F) Robustness in decoding across states shows a similar dependence on the correlation coefficient as for the model data (compare orange to Fig 5D, and red to Fig 6D), validating the analysis across real and model data. (G) Robustness across states of the cortical neurons exhibits a shape closer to the adaptive threshold model, characterized by a slower and later decay (for static decoding especially) than for the fixed threshold model (Fig 5E).