Dendritic Voltage Recordings Explain Paradoxical Synaptic Plasticity: A Modeling Study

Abstract

Experiments have shown that the same stimulation pattern that causes Long-Term Potentiation in proximal synapses, will induce Long-Term Depression in distal ones. In order to understand these, and other, surprising observations we use a phenomenological model of Hebbian plasticity at the location of the synapse. Our model describes the Hebbian condition of joint activity of pre- and postsynaptic neurons in a compact form as the interaction of the glutamate trace left by a presynaptic spike with the time course of the postsynaptic voltage. Instead of simulating the voltage, we test the model using experimentally recorded dendritic voltage traces in hippocampus and neocortex. We find that the time course of the voltage in the neighborhood of a stimulated synapse is a reliable predictor of whether a stimulated synapse undergoes potentiation, depression, or no change. Our computational model can explain the existence of different -at first glance seemingly paradoxical- outcomes of synaptic potentiation and depression experiments depending on the dendritic location of the synapse and the frequency or timing of the stimulation.

Keywords: STDP; computational neuroscience; dendritic recordings; model; synaptic plasticity; voltage.

FIGURE 1

Voltage-dependent plasticity model. (A) The activity X(t) of the presynaptic neuron induces local dendritic voltage changes u(t) in the postsynaptic neuron. The change in synaptic strength (LTP or LTD induction) depends on the timing of the presynaptic spike, and the voltage close to the synapse. (B) The presynaptic spike X(t) leaves a trace $\overline{x}$(t) at the synapse. The voltage u is low-pass filtered with a time constant τ₊ (for the variable $\overline{\text{u}}$₊) or τ_– (for $\overline{\text{u}}$_–). The amount of LTP is proportional to $\overline{x}$ multiplied by $\overline{\text{u}}$₊, while $\overline{\text{u}}$₊ is above a threshold θ₊. (C) Similarly, the amount of LTD is proportional to $\overline{x}$ multiplied by $\overline{\text{u}}$_–, while $\overline{\text{u}}$_– is above a threshold θ_– which is lower than θ₊ and increases when LTP occurs. (D–G) Plasticity in hippocampal model cells: extracellular afferent stimulation is paired with voltage-clamp of the postsynaptic neuron at different potentials (see Ngezahayo et al., 2000). 100 brief extracellular afferent stimulations are done at 2 different frequencies: 2 (full line) and 40 Hz (dotted line). (E,F) Synaptic strength w in percentage of its initial value as a function of voltage with respect to resting potential, in mV, for two sets of parameters. (G) The presynaptic trace $\overline{x}$ (blue) and the voltage u superimposed with its filtered versions $\overline{\text{u}}$₊ and $\overline{\text{u}}$_– (black) for 3 different values of clamped voltage (8, 20 and 30 mV) and different stimulation frequencies. The thresholds are indicated by the dashed green (θ₊) and dashed purple (θ_–) lines. The parameters are: (E,G) τ_x = 5ms, τ₊ = 6ms, τ_– = 15ms, θ₊ = 10mV, θ₀ = 5mV, A_LTP = 0.0001mV^{– 1}.ms^{– 1}, A_LTD = 0.0001mV^{– 1}.ms^{– 1}, b_θ = 31000mV.ms, τ_θ = 14ms. (F) τ_x = 5ms, τ₊ = 7ms, τ_– = 15ms, θ₊ = 13mV, θ₀ = 7mV, A_LTP = 0.0001mV^{– 1}.ms^{– 1}, A_LTD = 0.0001mV^{– 1}.ms^{– 1}, b_θ = 45000mV.ms, τ_θ = 5ms.

FIGURE 2

Subthreshold and spike-timing dependent plasticity at CA3 synapses in the hippocampus. (A) Experimental setup. Stimulation of CA3 recurrent inputs (blue electrode) was paired with a subthreshold stimulation of mossy fiber inputs (MF, brown electrode). The pairing is repeated 60 times at 0.1 Hz (Brandalise and Gerber, 2014). (B–D) Experimental voltage traces (black, middle panel) caused by subthreshold stimulations with a 10ms (B), 0 ms (C), or –40 ms (D) interval. Blue traces (top panels, $\overline{x}$(t)), green and purple (middle panels: $\overline{\text{u}}$₊ green full line; $\overline{\text{u}}$_– purple full line; θ₊ green dashed line; and θ_– purple dashed line), and black (bottom panels, w) show the time course of selected model variables during the simulated experiments. Two different cases are illustrated in (B), because for the same stimulation two types of voltage responses were recorded in the dendrite (black electrode in A): linear (left) and supralinear (right) ones. The supralinear responses correspond to the occurrence of dendritic spikes. (E) STDP protocol (Brandalise et al., 2016): stimulations of CA3 recurrent inputs were paired 50 times with brief somatic current injections (2 ms; 4 nA) which evoked action potentials (APs). (F) When 3 APs were evoked at a frequency of 200 Hz, dendritic spikes (black, right panels) occurred in 60% of the trials. In the remaining 40%, a linear response was generated (left); color of other traces as in B and C. (G) Responses were always linear, if only one AP was paired with stimulation of CA3 recurrent inputs. (B–D,F,G) Time of presynaptic stimulation is set to 0. (H) Plasticity outcome using different plasticity protocols. STDP protocol (see E): presynaptic stimulation paired either with 3 APs (see F), with 3 APs and concomitant application of a hyperpolarizing pulse, with 3 APs generated at a lower frequency (50 Hz) or with 1 AP (see G). Subthreshold protocol: CA3 stimulation only, no pairing; pairings with different time intervals (0 ms, 10 ms, –40 ms, see B–D) and, for the 10 ms time interval, simultaneously blocking the occurrence of supralinear events (10ms, block). Filled circles represent data from individual cells. Black error bars represent experimental mean ± SD. Red crosses represent simulations using the parameters obtained with the best fit. All data points in (H) are fitted with a single set of parameters (see text and Table 1). Differences in plasticity for the same protocol (e.g. 10 ms) arise due to differences in experimental voltage traces. Data shared by F. Brandalise. (I) Squared Error (SE) of the best fit subtracted from the SE obtained after increasing (upper panel) or decreasing (lower panel) each parameter by 5%, one parameter at a time (ΔSE).

FIGURE 3

Distance-dependent STDP at synapses between layer 2/3 and layer 5 pyramidal neurons in somatosensory cortex. (A–C) Voltage traces at distal (middle) and proximal (bottom) synapses. Postsynaptic bursts (3 action potentials, APs at 200 Hz) are paired with presynaptic action potentials (± 10 ms time interval, pairing frequency of 1 Hz, 150 repetitions). Experimental voltage traces u (black line) are redrawn from Letzkus et al. (2006). At distal synapses, dendritic spikes are generated. Glutamate trace $\overline{x}$ (blue line, top), filtered versions of the voltage $\overline{\text{u}}$₊ and $\overline{\text{u}}$_– (solid green and purple lines), and synaptic weight w (number: value of w after one pairing; dashed line: b_θ is set to 0, no veto) as a function of time. (A) During + 10 ms pairings, the value of the presynaptic trace $\overline{x}$ has already decreased significantly before $\overline{\text{u}}$₊ reaches θ₊. The amount of LTP is not high enough for the veto to have a significant impact on LTD induction. (B) In contrast, for –10 ms pairings, $\overline{x}$ switches from 0 to its maximal value 1 at around the time when $\overline{\text{u}}$₊ reaches its maximal value far above θ₊ and $\overline{\text{u}}$_– is slightly above θ_–. Therefore, the amount of LTP induced is high and significantly reduces LTD via an increase of the LTD threshold θ_–. (C) However, in the presence of NiCl₂ (blocks a subtype of voltage-gated calcium channels), the difference between $\overline{\text{u}}$₊ and θ₊ is significantly reduced compared to the control case in B, leading to no change in the synaptic strength. (D,E) Experimental and predicted change in synaptic weight. Crosses and dots or crosses with error bars represent plasticity from Letzkus et al. (2006) and red lines simulations. (D) Plasticity along the dendrite for the protocols described in A-C: 1pre-3 APs (left) or 3 APs-1 pre (right). EPSP rise time at the soma is a proxy of the distance between the plastic synapse and the soma (a distance of 110 μm, 330 μm and 660 μm correspond to a rise time of 1.8 ms, 2.5 ms and 3.5 ms, respectively, see Letzkus et al., 2006). See text for more details. (E) Plasticity at distal synapses in the presence of NiCl₂.

FIGURE 4

Pairing and timing-dependence of plasticity at neocortical synapses. (A) Two synaptically connected L5 neurons were stimulated with different time intervals (–10, 0, 10 and 25 ms) at different pairing repetition frequencies: 0.1 Hz, 10 Hz, 20 Hz and 40 Hz. (B) Simulated dendritic (black), somatic (orange) and experimentally recorded somatic (blue) voltage time course for + 10 ms time interval (number: peak value). The experimental voltage trace is redrawn from Sjöström et al. (2001); inset: EPSP time course. (C) Plasticity as a function of spike timing. Each panel represents one pairing repetition frequency. LTP is induced at high frequencies. Black errorbars represent data from Sjöström et al. (2001) and red squares represent our plasticity model. (D) Presynaptic trace $\overline{x}$ (blue), voltage u (black) and its filtered versions $\overline{\text{u}}$₊ (green) and $\overline{\text{u}}$_– (purple) for + 10 ms time interval: 10 Hz (left) or 40 Hz (right) repetition frequency.

FIGURE 5

Non-linear voltage-dependence. (A) The dendritic voltage is clamped for a fixed duration T and varying amplitudes Δu. The resulting squared voltage pulse is paired with a presynaptic spike X arriving 10 ms before the start of the pulse (dashed blue), 10 ms after the end of the pulse (dash-dotted blue), or in the center of the pulse (full blue). (B) Plasticity as a function of voltage amplitude Δu for T = 5 ms (left panels), T = 15 ms (middle panels) or T = 25 ms (right panels), using two sets of parameters (Letzkus: 100 or 10 pairings, top and middle panels respectively, or Brandalise: 10 pairings, bottom panels, see Table 1).