Potassium conductance dynamics confer robust spike-time precision in a neuromorphic model of the auditory brain stem

Abstract

A fundamental question in neuroscience is how neurons perform precise operations despite inherent variability. This question also applies to neuromorphic engineering, where low-power microchips emulate the brain using large populations of diverse silicon neurons. Biological neurons in the auditory pathway display precise spike timing, critical for sound localization and interpretation of complex waveforms such as speech, even though they are a heterogeneous population. Silicon neurons are also heterogeneous, due to a key design constraint in neuromorphic engineering: smaller transistors offer lower power consumption and more neurons per unit area of silicon, but also more variability between transistors and thus between silicon neurons. Utilizing this variability in a neuromorphic model of the auditory brain stem with 1,080 silicon neurons, we found that a low-voltage-activated potassium conductance (g(KL)) enables precise spike timing via two mechanisms: statically reducing the resting membrane time constant and dynamically suppressing late synaptic inputs. The relative contribution of these two mechanisms is unknown because blocking g(KL) in vitro eliminates dynamic adaptation but also lengthens the membrane time constant. We replaced g(KL) with a static leak in silico to recover the short membrane time constant and found that silicon neurons could mimic the spike-time precision of their biological counterparts, but only over a narrow range of stimulus intensities and biophysical parameters. The dynamics of g(KL) were required for precise spike timing robust to stimulus variation across a heterogeneous population of silicon neurons, thus explaining how neural and neuromorphic systems may perform precise operations despite inherent variability.

Keywords: bushy cells; gKL; heterogeneity; phase locking; silicon neuron.

Fig. 1.

Silicon bushy cell circuit and neuromorphic chip test setup. A: schematic of the analog complementary metal-oxide-semiconductor (CMOS) circuit that implements the bushy cell conductance equation. V_m, the analog membrane voltage, continuously drives the hyperpolarization-activated cation conductance (g_H), low-voltage-activated potassium conductance (g_KL), and sodium conductance (g_NA) circuits. REQ, the digital spike request, is acknowledged by ACK, which transiently drives the spike-resetting potassium conductance (g_KH) and temporarily inactivates g_NA. ANF_i, the auditory nerve inputs, are digital pulses, extended to 50 μs and low-pass filtered before driving the synaptic conductance (g_SYN). g_EXT, the externally controlled transistor, drives excitatory current-clamp stimulation. Currents produced by these conductances are integrated on V_int, which can be isolated from V_m by VCL, the voltage-clamp circuitry. The low-pass filter building block is outlined (LPF). B: the fabricated microchip consumes 3.5 mW of power and is housed on a custom-designed printed circuit board that supplies bias voltages to its circuits and facilitates real-time communication with test equipment.

Fig. 2.

Silicon bushy cell mimics several acoustic response features with g_KL intact or replaced by a static leak. A–C: dependence of 3 conductances on bushy cell membrane potential: g_KL (red), g_NA (blue), and g_H (green). Resting potential is indicated with an open triangle. Response to 10-ms depolarizing current pulse is shown at right. The software model's magnitudes and kinetics are based on in vitro measurements, averaged and scaled to account for physiological temperature (Rothman and Manis 2003c). Conductance axis range is 0 to 260 nS. Membrane voltage range is −70 to −35 mV. The silicon model is configured with g_KL intact or replaced by a static leak, implemented by reducing g_KL and converting g_H into a voltage-independent leak conductance. g_H did not impact the membrane's response to transient stimuli because it activates too slowly (time constant ∼50 ms) to dynamically affect spike-time precision during a 10-ms stimulus. In silico parameters are represented by currents. Conductanceaxis axis range is 0 to 150 nA. Membrane voltage range is 0 to 8 μA. D–F: peristimulus time histograms of instantaneous spike rate (0.1-ms bins) computed from 1,000 responses to a 25-ms pure tone at 60 dB SPL stimulus (left, 250 Hz; right, 8,000 Hz) for a software auditory nerve fiber and a silicon bushy cell with g_KL intact or replaced by a static leak. Vertical scale bar applies to all histograms. G: maximum vector strength across a range of pure tone frequencies.

Fig. 3.

Silicon bushy cell mimics intensity-invariant spike-time precision and spike-count reliability with g_KL intact, but not with a static leak. A: spike rate, vector strength, and entrainment index vs. intensity of a 250-Hz pure tone stimulus. Each data point is computed from 1,000 simulated presentations of a 25-ms pure tone at each intensity level (every 5 dB). Results are shown for a silicon bushy cell with g_KL intact and replaced by a static leak as well as the low-, medium-, and high-spontaneous rate auditory nerve fibers (ANF: LSR, MSR, HSR) that drives it. Vector strength and entrainment index are computed only if 10 or more spikes or intervals occurred. B: spike rasters of the first 100 trials. Mixed ANF distribution includes 1 LSR, 1 MSR, and 9 HSR fibers. C: vector strength at 70 and 120 dB SPL of ANF (top), bushy cell with g_KL intact (middle), and bushy cell with g_KL replaced by a static leak (bottom) for different ANF input distributions (mixed or all from the same spontaneous rate category). Error bars indicate 99% confidence intervals for each vector strength estimate. *Significant difference (P < 0.01) in vector strength at 70 vs. 120 dB SPL (see materials and methods); n.s. indicates no significant difference.

Fig. 4.

g_KL's dynamics confer intensity-invariant spike-time precision only at low stimulus frequencies. A: range of vector strengths at different stimulus frequencies. Upper bound is the maximum across all intensities. Lower bound is the minimum across intensities 5 to 50 dB above the intensity yielding maximum vector strength. B–D: silicon neuron's membrane responses to a single presentation of a 25-ms, 250-Hz pure tone at 120 dB SPL (top). B: spike times of all 11 ANF inputs during a single trial (trial 2 from Fig. 3B). C: membrane potential with g_KL intact (red) or replaced by a static leak (blue). D: instantaneous value of g_KL conductance.

Fig. 5.

g_KL's dynamics are necessary for intensity-invariant phase locking to amplitude-modulated high-frequency tones that contain periodic silence. Simulated and in silico responses of high best-frequency auditory neurons to 1,000 repetitions of a 25-ms, 8,000-Hz pure tone, 100% modulated by a 250-Hz half-wave-rectified sinusoid. A: driven spike rate and vector strength as in Fig. 3A, except spike phases are computed with respect to the envelope rather than the tone. B and C: stimulus waveforms and spike rate histograms during 2 modulation cycles (12–20 ms after stimulus onset). Vertical scale bar applies to all histograms.

Fig. 6.

g_KL's dynamics imbue intensity-invariant spike-time precision despite biophysical heterogeneity. A and B: parameter dependence (see key) of silicon bushy cell's spike rate and vector strength with g_KL intact (A) or replaced by a static leak (B). Auditory nerve data are provided for comparison (open circles). The reference model is driven by 16 fibers (4 LSR, 4 MSR, and 8 HSR); vector strength is computed only if 10 or more spikes occurred.

Fig. 7.

g_KL's dynamics protect against heterogeneity in a population of 1,080 silicon bushy cells. A: conductance-voltage curves' range of variation. Shaded region represents mean ± 2 SD; continuous line is from a silicon bushy cell next to the one shown in Fig. 2, B and C. B and C: plot of each cell's vector strength at 80 dB SPL vs. its vector strength at 120 dB SPL, with g_KL intact (B) or replaced by a static leak (C). Respective distributions of vector strengths (top and right histograms) and driven rates at 80 dB SPL (insets) are shown, color-coded on the basis of driven rate. Stimulus and model parameters are identical to those in Fig. 3. Values for the silicon neuron from Figs. 2–6 are indicated with triangles (rate) and squares (vector strength). Vector strengths from the software bushy cell with g_KL intact (as in Fig. 2A) or fixed at its steady-state level are indicated with circles.