Axonal noise as a source of synaptic variability

Abstract

Post-synaptic potential (PSP) variability is typically attributed to mechanisms inside synapses, yet recent advances in experimental methods and biophysical understanding have led us to reconsider the role of axons as highly reliable transmission channels. We show that in many thin axons of our brain, the action potential (AP) waveform and thus the Ca++ signal controlling vesicle release at synapses will be significantly affected by the inherent variability of ion channel gating. We investigate how and to what extent fluctuations in the AP waveform explain observed PSP variability. Using both biophysical theory and stochastic simulations of central and peripheral nervous system axons from vertebrates and invertebrates, we show that channel noise in thin axons (<1 µm diameter) causes random fluctuations in AP waveforms. AP height and width, both experimentally characterised parameters of post-synaptic response amplitude, vary e.g. by up to 20 mV and 0.5 ms while a single AP propagates in C-fibre axons. We show how AP height and width variabilities increase with a ¾ power-law as diameter decreases and translate these fluctuations into post-synaptic response variability using biophysical data and models of synaptic transmission. We find for example that for mammalian unmyelinated axons with 0.2 µm diameter (matching cerebellar parallel fibres) axonal noise alone can explain half of the PSP variability in cerebellar synapses. We conclude that axonal variability may have considerable impact on synaptic response variability. Thus, in many experimental frameworks investigating synaptic transmission through paired-cell recordings or extracellular stimulation of presynaptic neurons, causes of variability may have been confounded. We thereby show how bottom-up aggregation of molecular noise sources contributes to our understanding of variability observed at higher levels of biological organisation.

Figure 1. Definitions and methods.

(A) We record membrane potential and ionic currents at regularly placed points along the axon, and compare the (B) shape of the AP waveforms (black) and the resting membrane potential (dashed red). (C) Action potential features are determined individually for each AP. The amplitude of the AP is defined as the maximum membrane potential. The width is the delay between the crossings of the mid-height level. (D) AP shape fluctuations. Due to channel noise, APs triggered in an identical fashion will have different shapes across trials. Here APs in a 0.2 µm diameter axon from 5 trials out of 250 are superimposed. The point at which the membrane potential crossed the half-height line is used to align APs.

Figure 2. Random variability of AP waveform in thin axons (d = 0.2 µm) of four different types.

Subfigures display data for N>200 single APs triggered by identical stimuli and initial conditions (thick circle, 1×SD; dotted circle 3×SD). (A) Distribution of AP width and (B) AP height for rat hippocampal interneuron model axon. (C) Distribution of AP width and (D) AP height for C-fibre axons. (E) Distribution of AP width and (F) AP height for squid giant axons with Patlak channels. (G) Distribution of AP width and (H) AP height in squid giant axons.

Figure 3. Random variability of AP waveform in thin squid giant axon type axons of 3 diameters.

The subfigures display data for N>200 single APs triggered by identical stimuli and initial conditions in thin squid giant axon model. (A) Distribution of AP width and (B) AP height (red circle, 1×SD; dotted circle 3×SD). (C) Mean waveform of the AP at the proximal site. (D) Pairwise difference between an AP's shape at the proximal and the distal location. The average difference is plotted in thick black, while the light grey shaded area represents the 3×SD range. Grey lines represent sample traces plotted individually. (E) Fluctuations around the mean pairwise difference. The average difference is plotted in thick black (0 by definition), while the light grey shaded area represents the 3×SD range. Grey lines represent sample traces plotted individually.

Figure 4. Travelling APs' waveform fluctuations scale with axonal diameter with an inverse power low.

(A) Typical shape of an action potential in the squid giant axon. (B) The variability in the waveform at each moment in time (N = 250). We define the variability as 3×SD of the membrane potential at each point in time. (C) Log-log plot of 3×SD of fluctuations in AP shape over diameter for the rat hippocampal interneuron. (D) Log-log plot of 3×SD of fluctuations in AP shape over diameter for a C-fibre axon. (E) Log-log plot of 3×SD of fluctuations in AP shape over diameter for a squid giant axon (Patlak channels). (F) Log-log plot of 3×SD of fluctuations in AP shape over diameter for a squid giant axon.

Figure 5. Variability of the AP width and height scales with axon diameter.

The power-law relationship is valid at both proximal (left-pointed triangle) and distal (right-pointed triangle) points in all four types of axons. (A) Log-log plot of CV of AP height over diameter for a rat hippocampal interneuron. (B) Log-log plot of CV of AP width over diameter for a rat hippocampal interneuron. (C) Log-log plot of CV of AP height over diameter for a c-fibre axon. (D) Log-log plot of CV of AP width over diameter for a c-fibre axon. (E) Log-log plot of CV of AP height over diameter for a squid giant axon with Patlak channels. (F) Log-log plot of CV of AP width over diameter for a squid giant axon with Patlak channels. (G) Log-log plot of CV of AP height over diameter for a squid giant axon. (H) Log-log plot of CV of AP width over diameter for a squid giant axon.

Figure 6. Ion channel and current fluctuations underlie waveform variability.

Ion channel and current fluctuations in an AP travelling along 0.2 µm diameter axons. All traces are aligned at the instant when the rising AP crosses half-peak. Data from the squid giant axon model is on the left, while the right plots show data from the rat hippocampal interneuron model. Shaded areas in A–D are the 3×SD envelope around the mean curve (dark curve). Light curves in A–D represent a sample of individual traces. (A) Membrane potential waveforms (B) Number of open Na⁺ (red) and K⁺ (blue) channels (C) Current flowing through Na+ (red) and K⁺ (blue) channels (D) Net membrane current (sum of Na⁺, K⁺ and leak) (E) SD of Na⁺ (red), K⁺ (blue) and net membrane (green) currents.

Figure 7. Variability in synaptic processes due to fluctuation in AP waveforms.

For all subfigures, the mean waveform is plotted in black, SD in red, and 1%–99% quantiles in light blue. (A) Waveforms of 2000 consecutive APs arriving at the terminal end of an axon of 0.2 µm diameter rat hippocampal interneuron model axon. (B) Ca⁺⁺ current resulting from the integration of the above AP waveforms into a model of a pre-synaptic Ca⁺⁺ channel (see text for details). The current is scaled because we are only interested in its waveform. (C) Intracellular Ca⁺⁺ concentration for a large CNS synapse obtained by scaling the Ca⁺⁺ current waveform, and lengthening it. (D) Time course of vesicle release rates computed for a model of a Calyx-of-Held type synapse (see text for details).

Figure 8. Calculated distribution of synaptic response variability.

(A) Distribution of peak vesicle release rate in a large CNS synapse resulting from variability in (N = 2000) AP waveforms (see text for details). (B) Distribution of the total number of released vesicles in a model of Calyx-of-Held type synapse for (N = 2000) AP waveforms.

Figure 9. Allosteric model for the instantaneous rate of vesicle release in a Calyx of Held.

This model is used to estimate the vesicle release rate as a function of local transient Ca⁺⁺ concentration. Transition rates between states depend on Ca⁺⁺ local concentration in the vicinity of the vesicles. The release probability is higher the more Ca⁺⁺ ions are bound to the vesicle, according to a factor f = 31.3. The base rate constant I₊ is set to 2×10⁻⁴ s⁻¹.