Table 1.
Parameter values for the fractional leaky integrate-and-fire model.
Figure 1.
Comparison between the classical and fractional leaky integrate-and-fire models.
(A) Schematic circuit diagrams for the classical (left) and fractional order (right) leaky integrate-and-fire models. (B) Sub-threshold response in the classical (left) and fractional models (right). Both stimulated with nA. (C) The sub-threshold voltage response converges to a power-law function when
decreases. (D) While the classical model (left) generates regular spiking to a constant input, the fractional model (right) shows first spike latency and spike adaptation. Both models stimulated with
nA. (E) The first-spike latency produced by the fractional model becomes longer when
is smaller. (F) The inter-spike interval histogram as a function of
. The histogram shows power-law distribution as
. (G) The inter-spike intervals decrease over time as a function of
. The color key in C applies to F and G.
Figure 2.
The mean and instantaneous firing rate responses of the fractional model to constant input.
(A) Two levels of hyper-polarizing current followed by the same depolarizing current result in different spiking patterns. (B) The instantaneous firing rate against time for different conditions described in A. (C) The instantaneous firing rates against time for identical hyper-polarizing current (−3 nA) with different durations. (D–F) Comparison of mean and instantaneous firing rates. (D) Top: Applying a hyper-polarizing (pre-stimulus low, dashed lines) current before application of current steps to calculate firing rate responses. Bottom: as in the Top but applying a depolarizing (pre-stimulus high, solid lines) current. (E) The mean firing rates (mean FR) vs injected current show Type I response for both (pre-stimulus low) and (pre-stimulus high) current input paradigms described in D. The dashed and solid lines corresponding to pre-stimulus low and high, respectively, overlap. (F) The instantaneous firing rate to the stimulations described in D calculated from the first inter-spike interval depends on past activities. Dashed lines correspond to pre-stimulus low paradigm, solid lines correspond to pre-stimulus high.
Figure 3.
Sub-threshold and spiking fractional dynamics to oscillatory inputs.
(A–B) Voltage responses, impedances, and phase angles of the fractional order model in response to a ZAP current. (A) A time varying sub-threshold current input (Top) and the voltage responses for three different values of (Bottom). (B) Impedance and phase analysis for the simulations in A as a function of
. (C–D) Spiking response to just above threshold sinusoidal input (C, Top). The neuron generates an increasing number of spikes per cycle (C, Bottom). (D) The number of spikes per cycle for identical input as in C and varying
. (E–G) Response to supra-threshold sinusoidal input. (E) The fractional model with
instantaneous firing rate (black) in response to a sine wave input (blue). (F) The gain of the firing rate with respect to the period length of the input (Top) shows power-law dynamics when plotted in log-log (Bottom). The slope of the best-fit line (red) for the log-log gain curve is -
. (G) The phase lead of the firing rate in response to the same sine wave current. (H–J) Response to square periodic input. (H) In response to square wave current (Top), the fractional model displays upward and downward spike rate adaptation for
. (I) The instantaneous firing rate shows upward and downward adaptations in response to changes in the period of the square input (4, 8 and 16 s). (J) The time constants of both upward and downward adaptations in (I) increase when the period of the alternating input current increases. (K) The spike rate adaptation of L2/3 neocortical pyramidal neurons with period 16 s (Fig. 1C in [37]) is fitted with the spike rate adaptation of the fractional model with
(95% confidence interval) using least-squares fitting. The alternating input current is switched between 3.4 and 4 nA. For E–K we used
ms and
ms to better replicate the experimental data.
Figure 4.
Inter-spike interval adaptation and history dependence.
(A) Left: The spiking activity of the fractional model with a step current of 4 nA. Right: The inter-spike interval (ISI) curve of Layer 5 pyramidal neurons in primary motor cortex (Fig. 2B in [38]) is fitted with the ISI curve of the fractional model with (95% confidence interval with a least-squares fit). The first 7 ISIs of the model are removed for best fit. (B–C) Modeling the intrinsic memory of adapting pyramidal cells. (B) Left: The voltage trace of the model in response to a step current separated by 0.1 s inter-stimulus interval. Right: The ISIs of Cycle 1 and 5 as a function of ISI number. (C) The same as panel B, but with longer inter-stimulus interval, 2 s. For A–C
. (D) Memory induced pauses of the model with
depend on the magnitude of the current pulse. Left: Voltage traces with shorter and longer pauses in response to 1 nA and 4 nA current pulses, respectively. Right: The pause of the spiking activity increases as a function of the magnitude of the current pulse. For all we used
ms and
ms to better replicate the experimental data.
Figure 5.
Firing rate adaptation to changes in input variance.
(A) Top: Noisy input current with two standard deviations: 6+
nA,
= 1 or 2 nA. Bottom: The firing rate to noisy input current calculated from 100 trials,
. (B) Top: Time varying noisy input current.
5+
nA,
= 1, 4, 2, 1, 2, 1, 4 and 2 nA, consecutively, and the noise
is filtered with an alpha function
with
ms. Bottom: Instantaneous firing rate in response to the input current for
(blue) and
(black). For both
ms and
ms.
Figure 7.
The properties of the voltage-memory trace.
(A–D)The changing response of the memory trace across multiple spikes, = 0.2. (A) Voltage trace of the fractional model stimulated with
nA. Spikes have been clipped to emphasize the sub-threshold dynamics. (B)
for the data in A. (C) The memory trace for the data in A. (D) Overlapped memory traces for different inter-spike intervals during the same simulation. (E–G) The dynamics of the weight of the voltage-memory trace
and the fractional coefficient depend on the fractional exponent
. (E) When
decreases the weights increase. (F) The value of the weight
as a function of
and time. (G) The fractional coefficient of the Markov process
increases when
is decreased.
Figure 6.
Spike-time reliability increases as the fractional exponent decreases.
(A) Raster plots of the response of the fractional model to a noisy input under three different values of . (B) Spike-time reliability of the fractional model increases as the fractional exponent
decreases. (C) Reliability increases when the standard deviation of an embedded fixed signal increases. See Text and Methods for details.
Figure 8.
The memory trace dominates the fractional dynamics for low values of .
Markov term versus memory trace as a function of . The fractional model was stimulated with constant current (0.3 nA) for 5 seconds. For
the memory trace is zero and the voltage only moves along the y-axis.
Figure 9.
The fractional model and its analytic solution with memory reset show no spike adaptation.
(A) The spike train produced by the analytic solution with memory reset displays regular spiking. (B) The spike train produced with the fractional model with memory reset also displays regular spiking. (C) The spike train produced by the full fractional model without any memory reset display spike adaptation. (D). The firing rate curves of the analytic solution, fractional model with memory reset and full fractional model. For all panels = 0.1,
nA,
ms and
ms.