Home | About | Journals | Submit | Contact Us | Français |

**|**HHS Author Manuscripts**|**PMC2917246

Formats

Article sections

Authors

Related links

J Neurosci. Author manuscript; available in PMC 2010 December 1.

Published in final edited form as:

PMCID: PMC2917246

NIHMSID: NIHMS216055

Robert Legenstein,^{1,}^{*} Steven M. Chase,^{2,}^{3,}^{4} Andrew B. Schwartz,^{2,}^{3} and Wolfgang Maass^{1}

The publisher's final edited version of this article is available free at J Neurosci

See other articles in PMC that cite the published article.

It has recently been shown in a brain-computer interface experiment that motor cortical neurons change their tuning properties selectively to compensate for errors induced by displaced decoding parameters. In particular, it was shown that the 3D tuning curves of neurons whose decoding parameters were re-assigned changed more than those of neurons whose decoding parameters had not been re-assigned. In this article, we propose a simple learning rule that can reproduce this effect. Our learning rule uses Hebbian weight updates driven by a global reward signal and neuronal noise. In contrast to most previously proposed learning rules, this approach does not require extrinsic information to separate noise from signal. The learning rule is able to optimize the performance of a model system within biologically realistic periods of time under high noise levels. Furthermore, when the model parameters are matched to data recorded during the brain-computer interface learning experiments described above, the model produces learning effects strikingly similar to those found in the experiments.

Recent advances in microelectrode recording technology make it possible to sample the neural network generating behavioral output with brain-computer interfaces (BCIs). Monkeys using BCIs to control cursors or robotic arms improve with practice (Taylor et al., 2002; Carmena et al., 2003; Musallam et al., 2004; Schwartz, 2007; Ganguly and Carmena, 2009), indicating that learning-related changes are funneling through the set of neurons being recorded. In a recent report (Jarosiewicz et al., 2008), adaptation-related changes in neural firing rates were systematically studied in a series of BCI experiments. In that work, monkeys used motor cortical activity to control a cursor in a 3D virtual reality environment during a center-out movement task. When the activity of a subset of neurons was decoded incorrectly, to produce cursor movement at an angle to the intended movement, the tuning properties of that subset changed significantly more than for the subset of neurons for which activity was decoded correctly. This experiment demonstrated that motor cortical neurons may be able to solve the “credit assignment” problem: using only the global feedback of cursor movement, the subset of cells contributing more to cursor error underwent larger tuning changes. This adaptation strategy is quite surprising, since it is not clear how a learning mechanism is able to determine which subset of neurons needs to be changed.

In this article, we propose a simple biologically plausible reinforcement learning rule and apply it to a simulated 3D reaching task similar to the task in (Jarosiewicz et al., 2008). This learning rule is reward-modulated Hebbian: weight changes at synapses are driven by the correlation between a global reward signal, presynaptic activity, and the difference of the postsynaptic potential from its recent mean (Loewenstein and Seung, 2006). An important feature of the learning rule proposed in this article is that noisy neuronal output is used for exploration to improve performance. We show that large amounts of noise are beneficial for the adaptation process but not problematic for the readout system. In contrast to most other proposed reward-modulated learning rules, the version of the reward-modulated Hebbian learning rule that we propose does not require any external information to differentiate internal noise from synaptic input. We demonstrate that this learning rule is capable of optimizing performance in a neural network engaging in a simulated 3D reaching task. Furthermore, when compared to the results of Jarosiewicz et al. (2008), the simulation matches the differential-learning effects they report. Thus, this study shows that noise-driven learning can explain detailed experimental results about neuronal tuning changes in a motor control task and suggests that the corresponding reward-modulation of the learning process acts on specific sub-populations as an essential cortical mechanism for the acquisition of goal-directed behavior.

We consider several of the sections within Methods, below, to be crucial to understanding the results.

The methods are structured as follows. The experiments of Jarosiewicz et al. (2008) are briefly summarized in the following section, *Experiment: Learning effects in monkey motor cortex*. Simulation methods that we consider to be crucial to understanding the Results are prefaced in the subsection headings below with ‘*Simulation:*’. Simulation details that are needed for completeness, but not necessary for all readers are prefaced below with ‘*Simulation details:*’.

This section briefly describes the experiments of Jarosiewicz et al. (2008); a more complete description can be found in the original work. To extract intended movement from recorded neuronal activity in motor cortex, the firing rate of each neuron was fit as a function of movement direction using a cosine tuning curve (Georgopoulos et al., 1986; Schwartz, 2007). The preferred direction (PD) of the neuron was defined as the direction in which the cosine fit to its firing rate was maximal, and the modulation depth was defined as the difference in firing rate between the maximum of the cosine fit and the baseline (mean). The monkey’s intended movement velocity was extracted from the firing rates of a group of recorded units by computing the weighted sum of their PDs, where each weight was the unit’s normalized firing rate, i.e., by the population vector algorithm (Georgopoulos et al., 1988). (Note that units represented either well-isolated single neurons or a small number of neurons that could not be reliably distinguished, but were nevertheless tuned to movement as a group.)

In the learning experiments, the monkey controlled a cursor in a 3D virtual reality environment. The task for the monkey was to move the cursor from the center of an imaginary cube to a target appearing at one of its corners. Each of the experiments consisted of a sequence of four brain control sessions: *Calibration, Control, Perturbation*, and *Washout*. The tuning functions of an average of 40 recorded units were first obtained in the *Calibration* session, where an iterative procedure was used to obtain data for the linear regressions. These initial estimates of the PDs were later used for decoding neural trajectories into cursor movements. In order to distinguish between measured PDs and PDs used for decoding, we refer to the latter as “decoding PDs”, or dPDs. In the *Control*, *Perturbation*, and *Washout* sessions, the monkey had to perform a cursor control task in a 3D virtual reality environment (see Figure 1A). The cursor was initially positioned in the center of an imaginary cube; a target position on one of the corners of the cube was then randomly selected and made visible. When the monkey managed to hit the target position with the cursor (success), or a 3s time period expired (failure), the cursor position was reset to the origin and a new target position was randomly selected from the eight corners of the imaginary cube. In the *Control* session, the PDs measured during the *Calibration* session were used as dPDs for cursor control. In the *Perturbation* session, the dPDs of a randomly selected subset of units (25% or 50% of the recorded units) were altered from their control values by rotating them 90 degrees around one of the x, y, or z axes (all PDs were rotated around a common axis in each experiment). In this article, we term these units *rotated* units. The other dPDs remained the same as in the *Control* session. We term these units *non-rotated* units. In the subsequent *Washout* session, the measured PDs were again used for cursor control.

Description of the 3D cursor control task and network model for cursor control. A) The task was to move the cursor from the center of an imaginary cube to one of its eight corners. The target direction **y***(t) was given by the direction of the straight **...**

In the *Perturbation* session, the firing behavior of the recorded units changed to compensate for the altered dPDs. The authors observed differential effects of learning between the non-rotated and rotated groups of units. Rotated units tended to shift their PDs in the direction of dPD rotation, hence they compensated for the perturbation. The change of the PDs of non-rotated units was weaker and significantly less strongly biased towards the direction of rotation than the PDs of rotated units. We refer to this differential behavior of rotated and non-rotated units as the “credit assignment effect”.

Our aim was to explain the experimentally observed learning effects in the simplest possible model. This network model consisted of two populations of neurons connected in a feed-forward manner, see Figure 1B. The first population modeled those neurons that provide input to the neurons in motor cortex. It consisted of *m* = 100 neurons with activities *x _{1}(t),…,x_{m}(t)* . The second population modeled neurons in motor cortex that receive inputs from the input population. It consisted of

In monkeys, the transformation from motor cortical activity to arm movements involves a complicated system of several synaptic stages. In our model, we treated this transformation as a black box. Experimental findings suggest that monkey arm movements can be predicted quite well by a linear model based on the activities of a small number of motor cortex neurons (e.g., (Georgopoulos et al., 1989; Velliste et al., 2008). We therefore assumed that the direction of the monkey arm movement * y^{arm}(t)* at time

$${\mathit{y}}^{\mathit{\text{arm}}}(t)\propto {\displaystyle \sum _{i=1}^{\mathit{\text{n\_total}}}{S}_{i}(t){\mathit{q}}_{i}},$$

(1)

where **q**_{i}^{3} is the direction in which neuron *i* contributes to the movement. The vectors * q_{i}* were chosen randomly from a uniform distribution on the unit sphere (see

With the transformation from motor cortical neurons to monkey arm movements being defined, the input to the network for a given desired movement direction ** y^{*}** should be chosen such that motor cortical neurons produce a monkey arm movement close to

The total synaptic input *a _{i}(t)* to neuron

$${a}_{i}\left(t\right)={\displaystyle \sum _{j=1}^{m}{w}_{\mathit{\text{ij}}}{x}_{j}\left(t\right)+{\xi}_{i}\left(t\right)},\text{\hspace{1em}\hspace{1em}}{\xi}_{i}\mathit{\left(}t\mathit{\right)}\phantom{\rule{thinmathspace}{0ex}}\text{drawn from distribution}\phantom{\rule{thinmathspace}{0ex}}D\mathit{(}\nu \mathit{)},$$

(2)

where *w _{ij}* is the synaptic efficacy from input neuron

The activity *s _{i}(t)* of neuron

$${s}_{i}\left(t\right)=\sigma \left({a}_{i}\left(t\right)\right)$$

(3)

where σ : → is the threshold linear activation function which assures non-negative activities

$$\sigma (x)=\{\begin{array}{cc}\hfill x,\hfill & \hfill \mathit{\text{if}}\phantom{\rule{thinmathspace}{0ex}}x>0\hfill \\ \hfill 0,\hfill & \hfill \mathit{\text{otherwise}}\hfill \end{array}.$$

We modeled the cursor control task as shown in Figure 1A. Eight possible cursor target positions were located at the corners of a unit cube in 3D space with its center at the origin of the coordinate system. We simulated the closed-loop situation where the cursor moves according to the network output during simulated sessions and weights are adapted online. Before the simulation of a cursor control session, we determined the preferred directions * p_{i}* of simulated recorded neurons

Each simulated session consisted of a sequence of movements from the center to a target position at one of the corners of the imaginary cube, with online weight updates during the movements. To start a trial, the cursor position was initialized at the origin of the coordinate system and a target location was drawn randomly and uniformly from the corners of a cube with unit side length. The target location was held constant until the cursor hit the target, at which point the cursor was reset to the origin, a new target location was drawn, and another trial was simulated.

Each trial was simulated in the following way. At each time step *t* we performed a series of six computations. (1) The desired direction of cursor movement * y^{*}(t)* was computed as the difference between the target position

- 0)Initialize cursor to origin and pick target.
- 1)Compute desired direction
.**y**^{*}(t) - 2)Determine input activities
*x*._{1}(t),…,x_{m}(t) - 3)Determine motor cortical activities
*s*._{1}(t),…,s_{n_total}(t) - 4)Determine new cursor location
.**l**(t) - 5)Update synaptic weights
*w*._{ij} - 6)If target is not hit, set t to t+Δt and return to step 1.

To draw the contributions of simulated motor-cortical neurons to monkey arm movement * q_{i}=(q_{i1}, q_{i2}, q_{i3})^{T}* for

$${q}_{i1}=\sqrt{1-{q}_{i3}^{2}}\text{cos}({\phi}_{i}),\text{\hspace{1em}\hspace{1em}}{q}_{i2}=\sqrt{1-{q}_{i3}^{2}}\text{sin}({\phi}_{i}).$$

The activities of the neurons in the input population * x(t) = (x_{1}(t),…,x_{m}(t))^{T}* were determined such that the arm movement

$$\mathit{x}(t)={c}_{\mathit{\text{rate}}}{({W}^{\mathit{\text{total}}})}^{\u2020}{\mathit{Q}}^{\u2020}{\mathit{y}}^{*}(t),$$

(4)

where we used the scaling factor *c _{rate}* to scale the input activity such that the activities of the neurons in the simulated motor cortex could directly be interpreted as rates in Hz, i.e., such that their outputs were in the range between 0 and 120.

Note that this mapping was defined initially and kept fixed during each simulation. Thus, when *W ^{total}* was adapted by some learning rule, we still used the initial weights in the computation of the inputs since we assumed that the coding of desired directions did not change in the input coding.

As described above, a subset of the motor cortex population was chosen to model the recorded neurons that were used for cursor control. For each modeled recorded neuron *i* (Mercanzini et al., 2007), we determined the preferred direction **p**_{i}^{3}, baseline activity β_{i}, and modulation depth α_{i} as follows. We defined eight unit norm target directions * y*(1),…,y*(8)* as the eight directions from the origin to the eight corners of an imaginary cube centered at the origin. The activations

$$\sum}_{j=1}^{8}{({s}_{i}(j)-{\mathit{v}}_{i}^{T}{\mathit{y}}^{*}(j)-{\beta}_{i})}^{2$$

with respect to the vector * v_{i}=(v_{i1},v_{i2},v_{i3})^{T}* and β

$${s}_{i}(j)\approx {\beta}_{i}+{\alpha}_{i}\frac{{\mathit{y}}^{*}{(j)}^{T}{\mathit{p}}_{i}}{\Vert {\mathit{p}}_{i}\Vert},\text{for all}\phantom{\rule{thinmathspace}{0ex}}j.$$

After training, we re-estimated the PDs and analyzed how they changed due to learning.

In the simulated *Perturbation* session, the decoding preferred directions *p’ _{i}* of a randomly chosen subset of 50% of the modeled recorded neurons were rotated around one of the x, y, or z axes (all PDs were rotated around a common axis in each experiment). The dPDs of the non-rotated neurons were left the same as their measured PDs. The dPDs were then used to determine the movement velocity of the cursor as in Jarosiewicz et al. (2008) by the population vector algorithm (Georgopoulos et al., 1988): The cursor velocity was computed as the vector sum of the dPDs weighted by the corresponding normalized activities:

$$y\left(t\right)={k}_{s}\frac{d}{n}{\displaystyle \sum _{i=1}^{n}\frac{{s}_{i}\left(t\right)-{\beta}_{i}}{{\alpha}_{i}}}{\mathit{p}\text{'}}_{i},$$

(5)

where *d* is the movement dimensionality (in our case 3), *n* is the number of recorded neurons, and the constant *k _{s}* converts the magnitude of the population vector to speed. To set this speed factor in accordance with the experimental setup, we had to take the following temporal and geometrical considerations into account. In Jarosiewicz et al. (2008), the cursor position was updated every 30Hz. Hence, a time step in our simulation corresponded to 1/30 seconds in biological time. The imaginary cube in Jarosiewicz et al. (2008) had a side length of 11cm, whereas we used a cube with unit side length in our simulations. We therefore used a speed factor of

Finally, this velocity signal was integrated to obtain the cursor position **l**(t)

$$\mathbf{l}\left(t+\mathrm{\Delta}t\right)=\mathbf{l}\left(t\right)+\mathrm{\Delta}t\mathbf{y}\left(t\right),$$

where Δt = 1 in our simulations.

In order to simulate the experiments as closely as possible, we fit the noise in our model to the experimental data. To obtain quantitative estimates for the variability of neuronal responses in a cursor control task, we analyzed the 12 cursor control experiments in (Jarosiewicz et al., 2008) in which 50% of the neurons were perturbed (990 presented targets in total). We calculated for each recorded neuron the mean and variance of its firing rate over all successful trajectories with a common target. The firing rate was computed in a 200msec window half way to the target. This resulted in a total of 3592 unit-target location pairs. To smooth the data, running averages were taken of the sorted mean activities r and variances v:

$$\begin{array}{c}{\overline{r}}_{i}=\frac{1}{\tau +1}{\displaystyle {\sum}_{k=i}^{i+\tau}{r}_{k}}\hfill \\ {\overline{v}}_{i}=\frac{1}{\tau +1}{\displaystyle {\sum}_{k=i}^{i+\tau}{v}_{k}}\hfill \end{array}$$

We used a smoothing window of τ=*10*. Mean rates varied between 0 and 120Hz with a roughly exponential distribution such that mean rates above 60 Hz were very rare. In Figure 2, the smoothed variances were plotted as a function of the smoothed means . Since some recorded units can represent the activity of several neurons, this procedure may overestimate the amount of variability. This analysis was done on data from trained monkeys which have fairly stable movement trajectories. We thus obtained an estimate of neuronal firing rate variability for a given target direction.

Experimentally observed variances of the spike count in a 200 msec window for motor cortex neurons and noise in the neuron model. For a given target direction, the variance of the spike count for a unit scales approximately linearly with the mean activity **...**

The variance of the spike counts scaled roughly linearly with the mean spike count of a neuron for a given target location. This behavior can be obtained in our neuron model with noise that is a mixture of an activation-independent noise source and a noise source where the variance scales linearly with the noiseless activity of the neuron. In particular, the noise term ξ* _{i}(t)* of neuron

$${\nu}_{i}\left(x\left(t\right)\right)=\nu \left(1+\sqrt{\kappa \sigma \left({\displaystyle \sum _{j=1}^{m}{w}_{\mathit{\text{ij}}}{x}_{j}\left(t\right)}\right)}\right),$$

(6)

Recall that the input activities *x _{j}(t)* were scaled in such a way that the output of the neuron at time

Having estimated the variability of neuronal response, the learning rate η (see Eqn. 7 in Results) remained the last free parameter of the model. No direct experimental evidence for the value of η exists. We have therefore chosen the value of η such that after 320 target presentations the performance in the 25% perturbation task approximately matched the monkey performance. In Jarosiewicz et al. (2008) the performance was measured as the deviation of the cursor trajectory from the ideal straight line measured when the trajectory was half way to the target. In the 25% perturbation experiment, the monkey performance after 320 target presentations was approximately 3.2mm. By constraining this parameter according to the experimental data, we ensured that the model did not depend on any free parameter. We note that with this learning rate, the performance of the model was superior to monkey performance in the 50% perturbation experiment (see below).

To compute the deviation of the trajectory from the ideal one, trajectories were first rotated into a common frame of reference as in (Jarosiewicz et al., 2008). In this reference frame, the target is positioned at *(1,0,0) ^{T}*, movement along the

We model the learning effects observed by Jarosiewicz et al. (2008) through adaptation at a single synaptic stage, from a set of hypothesized input neurons to our motor cortical neurons. Adaptation of these synaptic efficacies *w _{ij}* will be necessary if the actual decoding PDs

$${R}_{\mathit{\text{ang}}}\left(t\right)=\frac{\mathit{y}{\left(t\right)}^{T}{\mathit{y}}^{*}\left(t\right)}{\Vert \mathit{y}\left(t\right)\Vert}.$$

This measure has a value of 1 if the cursor moves exactly in the desired direction, it is 0 if the cursor moves perpendicular to the desired direction, and it is −1 if the cursor movement is in the opposite direction. The angular match *R _{ang}(t)* will be used as the reward signal for adaptation below. For desired directions

$${R}_{\mathit{\text{batch}}}=\frac{1}{T}{\displaystyle \sum _{t=1}^{T}{R}_{\mathit{\text{ang}}}\left(t\right)}$$

is maximized.

The plasticity model used in this article is based on the assumption that learning in motor cortex neurons has to rely on a single global scalar neuromodulatory signal which carries information about system performance. One way for a neuromodulatory signal to influence synaptic weight changes is by gating local plasticity. In (Loewenstein and Seung, 2006) this idea was implemented by learning rules where the weight changes were proportional to the covariance between the reward signal *R* and some measure of neuronal activity *N* at the synapse, where *N* could correspond to the presynaptic activity, the postsynaptic activity, or the product of both. The authors showed that such learning rules can explain a phenomenon called Herrnstein’s matching law. Interestingly, for the analysis in (Loewenstein and Seung, 2006) the specific implementation of this correlation-based adaptation mechanism is not important. From this general class we investigate in this article the following learning rule:

$$\text{EH-rule}:\text{\hspace{1em}\hspace{1em}}\mathrm{\Delta}{w}_{\mathit{\text{ij}}}\left(t\right)=\eta \phantom{\rule{thinmathspace}{0ex}}{x}_{j}\left(t\right)\phantom{\rule{thinmathspace}{0ex}}\left[{a}_{i}\left(t\right)-{\overline{a}}_{i}\left(t\right)\right]\phantom{\rule{thinmathspace}{0ex}}\left[R\left(t\right)-\overline{R}\left(t\right)\right]$$

(7)

where (*t*) denotes the low-pass filtered version of some variable *z* with an exponential kernel; we used (*t*) = 0.8(*t* − 1) + 0.2*z*(*t*). We call this rule the exploratory Hebb rule (EH rule). The important feature of this learning rule is that apart from variables which are locally available for each neuron (*x _{j}*(

$$\mathrm{\Delta}{w}_{\mathit{\text{ij}}}\left(t\right)=\eta \phantom{\rule{thinmathspace}{0ex}}{x}_{j}\left(t\right)\phantom{\rule{thinmathspace}{0ex}}{\xi}_{i}\left(t\right)\phantom{\rule{thinmathspace}{0ex}}\left[R\left(t\right)-\overline{R}\left(t\right)\right].$$

This rule is a typical node-perturbation learning rule (Mazzoni et al., 1991; Williams, 1992; Baxter and Bartlett, 2001; Fiete and Seung, 2006) (see also the Discussion) that can be shown to approximate gradient ascent, see e.g. (Fiete and Seung, 2006). A simple derivation which shows the link between the EH rule and gradient ascent is given in the appendix.

The EH learning rule is different from other node-perturbation rules in one important aspect. In standard node-perturbation learning rules, the noise needs to be accessible to the learning mechanism separately from the output signal. For example, in (Mazzoni et al., 1991) and (Williams, 1992) binary neurons were used and the noise appears in the learning rule in the form of the probability of the neuron to output 1. In (Fiete and Seung, 2006), the noise term is directly incorporated in the learning rule. The EH rule instead does not directly need the noise signal, but a temporally filtered version of the activation of the neuron which is an estimate of the noise signal. Obviously, this estimate is only sufficiently accurate if the structure of the task is appropriate, i.e., if the input to the neuron is temporally stable on small time scales. We note that the filtering of postsynaptic activity makes the Hebbian part of the EH rule reminiscent of a linearized BCM rule (Bienenstock et al., 1982). The postsynaptic activity is compared with a threshold to decide whether the synapse is potentiated or depressed.

We simulated the two types of perturbation experiments reported in (Jarosiewicz et al., 2008) in our model network with 40 recorded neurons. In the first set of simulations, we chose 25% of the recorded neurons to be rotated neurons, and in the second set of simulations, we chose 50% of the recorded neurons to be rotated. In each simulation, 320 targets were presented to the model, which is similar to the number of target presentations in (Jarosiewicz et al., 2008). The performance improvement and PD shifts for one example run are shown in Figure 3. In order to simulate the experiments as closely as possible, we fit the noise and the learning rate in our model to the experimental data (see Methods). All neurons showed a tendency to compensate the perturbation by a shift of their PDs in the direction of the perturbation rotation. This tendency is stronger for rotated neurons. The training-induced shifts in PDs of the recorded neurons were compiled from 20 independent simulated experiments, and analyzed separately for rotated and non-rotated neurons. The results are in good agreement with the experimental data (Figure 4). In the simulated 25% perturbation experiment, the mean shift of the PD for rotated neurons was 8.2 ± 4.8 degrees, whereas for non-rotated neurons, it was 5.5 ± 1.6 degrees. This is a relatively small effect, similar to the effect observed in (Jarosiewicz et al., 2008) where the PD shifts were 9.86 degrees for rotated units and 5.25 degrees for non-rotated units. A stronger effect can be found in the 50% perturbation experiment (see below). We also compared the deviation of the trajectory from the ideal straight line in rotation direction half way to the target (see Methods) from early trials to the deviation of late trials. In early trials, the trajectory deviation was 9.2 ± 8.8 mm, which was reduced by learning to 2.4 ± 4.9 mm. In the simulated 50% perturbation experiment, the mean shift of the PD for rotated neurons was 18.1 ± 4.2 degrees, whereas for non-rotated neurons, it was 12.1 ± 2.6 degrees. Again, the PD shifts are very similar to those in the monkey experiments: 21.7 degrees for rotated units and 16.11 degrees for non-rotated units. The trajectory deviation was 23.1 ± 7.5 mm in early trials, and 4.8 ± 5.1 mm in late trials. Here, the early deviation was stronger than in the monkey experiment, while the late deviation was smaller.

One example simulation of the 50% perturbation experiment with the EH rule and data-derived network parameters. A) Angular match *R*_{ang} as a function of learning time. Every 100th time point is plotted. B) PD shifts projected onto the rotation plane (the **...**

PD shifts in simulated *Perturbation* sessions are in good agreement with experimental data (compare to Figure 3A,B in (Jarosiewicz et al., 2008)). Shift in the PDs measured after simulated perturbation sessions relative to initial PDs for all units in **...**

The EH rule falls into the general class of learning rules where the weight change is proportional to the covariance of the reward signal and some measure of neuronal activity (Loewenstein and Seung, 2006). Interestingly, the specific implementation of this idea influences the learning effects observed in our model. We performed the same experiment with slightly different correlation-based rules

$$\mathrm{\Delta}{w}_{\mathit{\text{ij}}}\left(t\right)=\eta \phantom{\rule{thinmathspace}{0ex}}{x}_{j}\left(t\right){a}_{i}\left(t\right)\left[R\left(t\right)-\overline{R}\left(t\right)\right],\text{and}$$

(8)

$$\mathrm{\Delta}{w}_{\mathit{\text{ij}}}\left(t\right)=\eta \phantom{\rule{thinmathspace}{0ex}}{x}_{j}\left(t\right)\left[{a}_{i}\left(t\right)-{\overline{a}}_{i}\left(t\right)\right]R\left(t\right),$$

(9)

where the filtered postsynaptic activation or the filtered reward was not taken into account. Compare these to the EH rule (Eqn. 7). These rules also converge with performance similar to the EH rule. However, no credit assignment effect can be observed with these rules. In the simulated 50% perturbation experiment, the mean shift of the PD of rotated neurons (non-rotated neurons) was 25.5 ± 4.0 (26.8 ± 2.8) degrees for rule (8) and 12.8 ± 3.6 (12.0 ± 2.4) degrees for rule (9), see Figure 5. Only when deviations of the reward from its local mean and deviations of the activation from its local mean are *both* taken into account do we observe differential changes in the two populations of cells.

PD shifts in simulated 50% *Perturbation* sessions with learning rules 8 and 9. Dots represent individual data points and black circled dots represent the means of the rotated (red) and non-rotated (blue) units. No credit assignment effect can be observed **...**

In the monkey experiment, training in the *Perturbation* session also resulted in a decrease of the modulation depth of rotated neurons, which led to a relative decrease of the contribution of these neurons to the cursor movement. A qualitatively similar result could be observed in our simulations. In the 25% perturbation simulation, modulation depths of rotated neurons changed on average by −2.7 ± 4.3 Hz, whereas modulation depths of non-rotated neurons changed on average by 2.2 ± 3.9 Hz (average over 20 independent simulations; a negative change indicates a decreased modulation depth in the *Perturbation* session relative to the *Control* session). In the 50% perturbation simulation, the changes in modulation depths were on average −3.6±5.5 Hz for rotated neurons and 5.4 ± 6.0 Hz for non-rotated neurons (when comparing these results to experimental results, one has to take into account that modulation depths in monkey experiments were around 10Hz, whereas in the simulations, they were around 25Hz). Thus, the relative contribution of rotated neurons on cursor movement decreased during the *Perturbation* session.

It was reported in (Jarosiewicz et al., 2008) that after the *Perturbation* session, PDs returned to their original values in a subsequent *Washout* session where the original PDs were used as decoding PDs. We simulated such *Washout* sessions after our simulated *Perturbation* sessions in the model and found a similar effect, see Figure 6A, B. However, the retuning in our simulation is slower than observed in the monkey experiments. In the experiments, it took about 160 target presentations until mean PD shifts relative to PDs in the *Control* session were around zero. This fast unlearning is consistent with the observation that adaptation and de-adaptation in motor cortex can occur at substantially different rates, likely reflecting two separate processes (Davidson and Wolpert, 2004). We did not model such separate processes, thus the time-scales for adaptation and de-adaptation are the same in the simulations. In a simulated *Washout* session with a larger learning rate, we found faster convergence of PDs to original values, see Figure 6C, D.

PDs shifts in simulated *Washout* sessions. Shift in the PDs (mean over 20 trials) for rotated neurons (gray) and non-rotated neurons (black) relative to PDs of the *Control* session as a function of the number of targets presented for 25% perturbation (left) **...**

The performance of the system before and after learning is shown in Figure 7. The neurons in the network after training are subject to the same amount of noise as the neurons in the network before training, but the angular match after training shows much less fluctuation than before training. We therefore conjectured that the network automatically suppresses jitter in the trajectory in the presence of high exploration levels *v*. We quantified this conjecture by computing the mean angle between the cursor velocity vector with and without noise for 50 randomly drawn noise samples. In the mean over the 20 simulations and 50 randomly drawn target directions, this angle was 10 ± 2.7 degrees (mean ± STD) before learning and 9.6 ± 2.5 degrees after learning. Although only a slight reduction, it was highly significant when the mean angles before and after learning were compared for identical target directions and noise realizations (p <0.0002, paired t-test). This is not an effect of increased network weights, because weights increased only slightly and the same test where weights were normalized to their initial L_{2} norm after training produced the same significance value.

Comparison of network performance before and after learning for 50% perturbation. Angular match *R*_{ang}(t) of the cursor movements in one reaching trial before (gray) and after (black) learning as a function of the time since the target was first made visible. **...**

Psychophysical studies in humans (Imamizu et al., 1995) and monkeys (Paz and Vaadia, 2005) showed that the learning of a new sensorimotor mapping generalizes poorly to untrained directions with better generalization for movements in directions close to the trained one. It was argued in (Imamizu et al., 1995) that this is evidence for a neural network-like model of sensorimotor mappings. The model studied in this article exhibits similar generalization behavior. When training is constrained to a single target location, performance is optimized in this direction while the performance clearly decreased as target direction increased from the trained angle, see Figure 8.

When we compare the results obtained by our simulations to those of monkey experiments (compare Figure 4 to Figure 3 in (Jarosiewicz et al., 2008)), it is interesting that quantitatively similar effects were obtained with noise levels that were measured in the experiments. We therefore explored whether the fitting of parameters to values extracted from experimental data was important by exploring the effect of different exploration levels and learning rates on performance and PD shifts.

The amount of noise was controlled by modifying the exploration level ν, see Eqn. 6. For some extreme parameter settings, the EH rule can lead to large weights. We therefore implemented a homeostatic mechanism by normalizing the weight vector of each neuron after each update, i.e., the weight after the *t*-th update step is given by

$${w}_{ij}\left(t+1\right)=\sqrt{{{\displaystyle \sum _{k}{w}_{ik}\left(t\right)}}^{2}}\frac{{w}_{ij}\left(t\right)+\mathrm{\Delta}{w}_{ij}\left(t\right)}{\sqrt{{{\displaystyle \sum _{k}\left({w}_{ik}\left(t\right)+\mathrm{\Delta}{w}_{ik}\left(t\right)\right)}}^{2}}}$$

Employing the EH learning rule, the network converged to weight settings with good performance for most parameter settings, except for large learning rates and very large noise levels. Note that good performance is achieved even for large exploration levels of ν≈*60*Hz, see Figure 9A. The good performance of the system shows that already a very small network can utilize large amounts of noise for learning while this noise does not interfere with performance.

Behavior of the EH rule in simulated *Perturbation* sessions (50% perturbed neurons) for different parameter settings. All plotted values are means over 10 independent simulations. Logarithms are to the basis of 2. The black circle indicates the parameter **...**

We investigated the influence of learning on the PDs of circuit neurons. The amount of exploration and the learning rate η both turned out be important parameters. The tuning changes reported in neurons of monkeys subsumed under the term ``credit assignment effect" were qualitatively met by our model networks for most parameter settings, see Figure 9, except for very large learning rates (when learning does not work) and very small learning rates, compare panels B and C. Quantitatively, the amount of PD shift especially for rotated neurons strongly depends on the exploration level, with shifts close to 50 degrees for large exploration levels.

To summarize, for small levels of exploration, PDs change only slightly and the difference in PD change between rotated and non-rotated neurons is small, while for large noise levels, PD change differences can be quite drastic. Also the learning rate η influences the amount of PD shifts. This shows that the learning rule guarantees good performance and a qualitative match to experimentally observed PD shifts for a wide range of parameters. However, for the quantitative fit found in our simulations, the parameters extracted from experimental data turned out to be important.

By implementing a learning rule that utilizes neuronal noise as an exploratory signal for parameter adaptation, we have successfully simulated experimental results showing selective learning within a population of cortical neurons (Jarosiewicz et al., 2008). This learning rule implements synaptic weight updates based on the instantaneous correlation between the deviation of a global error from its recent mean and the deviation of the neural activity from its recent mean; all of these parameters would be readily accessible in a biological system. Strikingly, it turns out that the use of noise levels similar to those that had been measured in experiments was essential to reproduce the learning effects found in the monkey experiments.

Jarosiewicz and colleagues (2008) discussed three possible strategies that could be used to compensate for the errors caused by the perturbations: re-aiming, re-weighting, and re-mapping. With re-aiming, the monkey would compensate for perturbations by aiming for a virtual target located in the direction that offsets the visuomotor rotation. The authors identified a global change in the measured PDs of all neurons, indicating that monkeys utilized a re-aiming strategy. Re-weighting would reduce the errors by selectively suppressing the use of rotated units, i.e. a reduction of their modulation depths relative to the modulation depths of non-rotated units. The same reduction was found in the firing rates of the rotated neurons in the data. A re-mapping strategy would selectively change the directional tunings of rotated units. As discussed above, rotated neurons shifted their PDs more than the non-rotated population. Hence, the authors found elements of all three strategies in their data. We identified in our model all three elements of neuronal adaptation, i.e. a global change in activity of neurons (all neurons changed their tuning properties; re-aiming), a reduction of modulation depths for rotated neurons (re-weighting), and a selective change of the directional tunings of rotated units (re-mapping). This modeling study therefore suggests that all three elements could be explained by a single learning mechanism. Furthermore, the credit assignment phenomenon observed by Jarosiewicz and colleagues (re-weighting and re-mapping) is an emergent feature of our learning rule.

Although the match of simulation results to experimental results is quite good, systematic differences exist. The change in simulated modulation depth was about twice that found in the experiments. It also turned out that the model produced smaller trajectory deviations after learning in the 50% deviation experiment. Such quantitative discrepancies could be attributed to the simplicity of the model. However, another factor that could systematically contribute to all of the stronger effects could be the accurate reward signal modeled at the synapse. We did not incorporate noisy reward signals in our model, however, because this would introduce a free parameter with no available evidence for its value. Instead, the parameters of the presented model were strongly constrained: the noise level was estimated from the data, and the learning rate was chosen such that the average trajectory error in the 25% perturbation experiment was comparable to that in experiments after a given number of trials.

Several reward-modulated Hebbian learning rules have been studied, both in the context of rate-based (Barto et al., 1983; Mazzoni et al., 1991; Williams, 1992; Baxter and Bartlett, 1999; Loewenstein and Seung, 2006) and spiking-based models (Xie and Seung, 2004; Fiete and Seung, 2006; Pfister et al., 2006; Baras and Meir, 2007; Farries and Fairhall, 2007; Florian, 2007; Izhikevich, 2007; Legenstein et al., 2008) They turn out to be viable learning mechanisms in many contexts and constitute a biologically plausible alternative to the backpropagation-based mechanisms preferentially used in artificial neural networks. Such three-factor learning rules are well studied in cortico-striatal synapses where the three factors are pre- and postsynaptic activity and dopamine, see e. g. (Reynolds and Wickens, 2002). The current conclusion drawn from the experimental literature is that pre- and postsynaptic activity is needed for plasticity induction. Depression is induced at low dopamine levels and potentiation is induced at high dopamine levels. The EH-rule is in principle consistent with these observations although it introduces the additional dependency on the recent postsynaptic rate and reward that have not been rigorously tested experimentally.

Reinforcement learning takes place when an agent learns to choose optimal actions based on some measure of performance. In order to improve performance, the agent has to explore different behaviors. In neuronal reinforcement learning systems, exploration is often implemented by some noise source that perturbs the operation in order to explore whether parameter settings should be adjusted to increase performance. In songbirds, syllable variability results in part from variations in the motor command, i.e. the variability of neuronal activity (Sober et al., 2008). It has been hypothesized that this motor variability reflects meaningful motor exploration that can support continuous learning (Tumer and Brainard, 2007). Two general classes of perturbation algorithms can be found in the literature. Either the tunable parameters of the system (weights) are perturbed (Jabri and Flower, 1992; Cauwenberghs, 1993; Seung, 2003) or the output of nodes in the network are perturbed (Mazzoni et al., 1991; Williams, 1992; Baxter and Bartlett, 2001; Fiete and Seung, 2006). The latter have the advantage that the perturbation search space is smaller and that the biological interpretation of the perturbation as an internal neural noise is more natural. Another interesting idea is the postulation of an ’experimenter’, that is, a system that injects noisy current into trained neurons. Some evidence for an experimenter exists in the song-learning system of zebra finches (Fiete et al., 2007). For the EH learning rule, the origin of the exploratory signal is not critical, as long as the trained neurons are noisy. The EH learning rule is in its structure similar to the rule proposed in (Fiete and Seung, 2006). However, while it had to be assumed in (Fiete and Seung, 2006) that the experimenter signal (ξ* _{i}(t)* in our notation) is explicitly available and distinguishable from the membrane potential at the synapse, the EH rule does not rely on this separation. Instead it exploits the temporal continuity of the task, estimating ξ

Often perturbation algorithms use eligibility traces in order to link perturbations at time *t* to rewards delivered at some later point in time *t´* > *t*. In fact, movement evaluation may be slow and the release/effect of neuromodulators may add to the delay in response imparted to neurons in the trained area. For simplicity, we did not use eligibility traces and assumed that evaluation by the critic can be done quite fast.

The EH rule falls into the general class of learning rules where the weight change is proportional to the covariance of the reward signal and some measure of neuronal activity (Loewenstein and Seung, 2006). Interestingly, the specific implementation of this idea influences the learning effects observed in our model. In particular, we found that the implementations given by rules (8) and (9) do not exhibit the reported credit assignment effect.

The results of this modeling paper also support the hypotheses introduced in (Rokni et al., 2007). The authors presented data suggesting that neural representations change randomly (background changes) even without obvious learning, while systematic task-correlated representational changes occur within a learning task. They proposed a theory based on three assumptions: (1) representations in motor cortex are redundant, (2) sensory feedback is translated to synaptic changes in a task, and (3) the plasticity mechanism is noisy. These assumptions are also met in our model of motor cortex learning. The authors also provided a simple neural network model where the stochasticity of plasticity was modeled directly by random weight changes. In our model, such stochasticity arises from the firing rate noise of the model neurons and it is necessary for task-dependent learning. This neuronal behavior together with the EH rule also leads to background synaptic changes in the absence of obvious learning (i. e. when performance is perfect or near-perfect).

Reward-modulated learning rules capture many of the empirical characteristics of local synaptic changes thought to generate goal-directed behavior based on global performance signals. The EH rule is one particularly simple instance of such a rule which emphasizes an exploration signal, a signal which would show up as “noise” in neuronal recordings. We showed that large exploration levels are beneficial for the learning mechanism without interfering with baseline performance, because of readout pooling effects. The study therefore provides a hypothesis about the role of “noise” or ongoing activity in cortical circuits as a source for exploration utilized by local learning rules. The data from (Jarosiewicz et al., 2008) suggest that the level of noise in motor cortex is quite high. Under such realistic noise conditions, our model produces effects strikingly similar to those found in the monkey experiments, which suggests this noise is essential for cortical plasticity. Obviously, these learning mechanisms are important for neural prosthetics, since they allow closed-loop corrections for poor extractions of movement intention. In addition, these learning mechanisms may be a general feature used for the acquisition of goal-directed behavior.

This work was supported by the Austrian Science Fund FWF [S9102-N13, to R.L. and W.M.]; the European Union [FP6-015879 (FACETS), FP7-506778 (PASCAL2), FP7-231267 (ORGANIC) to R.L. and W.M.]; and by the National Institutes of Health [R01-NS050256, EB005847, to A.B.S.].

In the following we give a simple derivation which shows that the EH rule performs gradient ascent on the reward signal *R(t)*. The weights should change in the direction of the gradient of the reward signal, which is given by the chain rule as

$$\frac{\partial R\left(t\right)}{\partial {w}_{\mathit{\text{ij}}}}=\frac{\partial R\left(t\right)}{\partial {a}_{i}\left(t\right)}\frac{\partial {a}_{i}\left(t\right)}{\partial {w}_{\mathit{\text{ij}}}}=\frac{\partial R\left(t\right)}{\partial {a}_{i}\left(t\right)}{x}_{j}\left(t\right),$$

(A1)

where *a _{j}(t)* is the total synaptic input to neuron

$$R\left(t\right)-{R}_{0}\left(t\right)\approx {\displaystyle \sum _{k}\frac{\partial R\left(t\right)}{\partial {a}_{k}\left(t\right)}{\xi}_{k}\left(t\right)}.$$

(A2)

Multiplying this equation with ξ* _{i}(t)* and averaging over different realizations of the noise, we obtain the correlation between the reward at time

$$\langle \left[R\left(t\right)-{R}_{0}\left(t\right)\right]{\xi}_{i}\left(t\right)\rangle \approx {\displaystyle \sum _{k}\frac{\partial R\left(t\right)}{\partial {a}_{k}\left(t\right)}\langle {\xi}_{k}\left(t\right){\xi}_{i}\left(t\right)\rangle}={\mu}^{2}\frac{\partial R\left(t\right)}{\partial {a}_{i}\left(t\right)}.$$

(A3)

The last equality follows from the assumption that the noise signal is temporally and spatially uncorrelated. Hence, the derivative of the reward signal with respect to the activation of neuron *i* is

$$\frac{\partial R\left(t\right)}{\partial {a}_{i}\left(t\right)}\approx \frac{1}{{\mu}^{2}}\langle \left[R\left(t\right)-{R}_{0}\left(t\right)\right]{\xi}_{i}\left(t\right)\rangle .$$

(A4)

Since ξ_{i}(*t*) = 0, we find

$${a}_{i}\left(t\right)-\langle {a}_{i}\left(t\right)\rangle ={\displaystyle \sum _{j=1}^{m}{w}_{\mathit{\text{ij}}}{x}_{j}\left(t\right)}+{\xi}_{i}\left(t\right)-{\displaystyle \sum _{j=1}^{m}{w}_{\mathit{\text{ij}}}{x}_{j}\left(t\right)}-\langle {\xi}_{i}\left(t\right)\rangle ={\xi}_{i}\left(t\right)$$

and we can write Eqn. A4 as

$$\frac{\partial R\left(t\right)}{\partial {a}_{i}\left(t\right)}\approx \frac{1}{{\mu}^{2}}\langle \left[R\left(t\right)-{R}_{0}\left(t\right)\right]\left[{a}_{i}\left(t\right)-\langle {a}_{i}\left(t\right)\rangle \right]\rangle .$$

(A5)

We note that

$$\langle \left[R\left(t\right)-{R}_{0}\left(t\right)\right]\left[{a}_{i}\left(t\right)-\langle {a}_{i}\left(t\right)\rangle \right]\rangle =\langle R\left(t\right){a}_{i}\left(t\right)\rangle -\langle R\left(t\right)\rangle \langle {a}_{i}\left(t\right)\rangle =\langle \left[R\left(t\right)-\langle R\left(t\right)\rangle \right]\left[{a}_{i}\left(t\right)-\langle {a}_{i}\left(t\right)\rangle \right]\rangle .$$

Using this result in Eqn. A1, we obtain

$$\frac{\partial R\left(t\right)}{\partial {w}_{\mathit{\text{ij}}}}\approx \frac{1}{{\mu}^{2}}\langle \left[R\left(t\right)-\langle R\left(t\right)\rangle \right]\left[{a}_{i}\left(t\right)-\langle {a}_{i}\left(t\right)\rangle \right]\rangle {x}_{j}\left(t\right).$$

(A6)

In our implementation, the EH learning rule estimates *a _{i}*(

- Baras D, Meir R. Reinforcement Learning, Spike-Time-Dependent Plasticity, and the BCM Rule. Neural Computation. 2007;19:2245–2279. [PubMed]
- Barto AG, Sutton RS, Anderson CW. Neuronlike adaptive elements that can solve difficult learning and control problems. IEEE transactions on systems, man, and cybernetics. 1983;13:835–846.
- Baxter J, Bartlett PL. Research School of Information Sciences and Engineering. Australian National University; 1999. Direct Gradient-Based Reinforcement Learning: I. Gradient Estimation Algorithms.
- Baxter J, Bartlett PL. Infinite-Horizon Policy-Gradient Estimation. Journal of Artificial Intelligence Research. 2001;15:319–350.
- Bienenstock EL, Cooper LN, Munro PW. Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex. J Neurosci. 1982;2:32–48. [PubMed]
- Carmena JM, Lebedev MA, Crist RE, O'Doherty JE, Santucci DM, Dimitrov DF, Patil PG, Henriquez CS, Nicolelis MA. Learning to control a brain-machine interface for reaching and grasping by primates. PLoS Biol. 2003;1 [PMC free article] [PubMed]
- Cauwenberghs G. A Fast Stochastic Error-Descent Algorithm for Supervised Learning and Optimization. In: Hanson SJe, Cowan JD, Giles CL., editors. Advances in Neural Information Processing Systems. San Mateo, CA: Morgan Kaufmann; 1993. pp. 244–251.
- Davidson PR, Wolpert DM. Scaling down motor memories: de-adaptation after motor learning. Neurosci Lett. 2004;370:102–107. [PubMed]
- Farries MA, Fairhall AL. Reinforcement Learning with Modulated Spike Timing-Dependent Synaptic Plasticity. J Neurophysiol. 2007;98:3648–3665. [PubMed]
- Fiete IR, Seung HS. Gradient Learning in Spiking Neural Networks by Dynamic Perturbation of Conductances. Physical Review Letters. 2006;97 048104-048101. [PubMed]
- Fiete IR, Fee MS, Seung HS. Model of birdsong learning based on gradient estimation by dynamic perturbation of neural conductances. J Neurophysiol. 2007;98:2038–2057. [PubMed]
- Florian RV. Reinforcement learning through modulation of spike-timing-dependent synaptic plasticity. Neural Computation. 2007;19:1468–1502. [PubMed]
- Ganguly K, Carmena JM. Emergence of a stable cortical map for neuroprosthetic control. PLoS Biol. 2009;7 e1000153. [PMC free article] [PubMed]
- Georgopoulos AP, Schwartz AP, Ketner RE. Neuronal population coding of movement direction. Science. 1986;233:1416–1419. [PubMed]
- Georgopoulos AP, Ketner RE, Schwartz AP. Primate motor cortex and free arm movements to visual targets in three- dimensional space. II. Coding of the direction of movement by a neuronal population. J Neurosci. 1988;8:2928–2937. [PubMed]
- Georgopoulos AP, Crutcher MD, Schwartz AB. Cognitive spatial-motor processes. 3. Motor cortical prediction of movement direction during an instructed delay period. Exp Brain Res. 1989;75:183–194. [PubMed]
- Imamizu H, Uno Y, Kawato M. Internal representations of the motor apparatus: implications from generalization in visuomotor learning. J Exp Psychol Hum Percept Perform. 1995;21:1174–1198. [PubMed]
- Izhikevich EM. Solving the Distal Reward Problem through Linkage of STDP and Dopamine Signaling. Cerebral Cortex. 2007;17:2443–2452. [PubMed]
- Jabri M, Flower B. Weight Perturbation: An Optimal Architecture and Learning Technique for Analog VLSI Feedforward and Recurrent Multilayer Networks. IEEE Transactions on Neural Networks. 1992;3:154–157. [PubMed]
- Jarosiewicz B, Chase SM, Fraser GW, Velliste M, Kass RE, Schwartz AB. Functional network reorganization during learning in a brain-computer interface paradigm. Proc Nat Acad Sci USA. 2008 [PubMed]
- Legenstein R, Pecevski D, Maass W. A Learning Theory for Reward-Modulated Spike-Timing-Dependent Plasticity with Application to Biofeedback. PLoS Computational Biology. 2008;4:1–27. [PMC free article] [PubMed]
- Loewenstein Y, Seung HS. Operant matching is a generic outcome of synaptic plasticity based on the covariance between reward and neural activity. Proc Nat Acad Sci USA. 2006;103:15224–15229. [PubMed]
- Mazzoni P, Andersen RA, Jordan MI. A more biologically plausible learning rule for neural networks. Proc Nat Acad Sci USA. 1991;88:4433–4437. [PubMed]
- Mercanzini A, Reddy S, Velluto D, Colin P, Maillard A, Bensadoun JC, Bertsch A, Hubbell JA, Renaud P. Controlled release drug coatings on flexible neural probes; Conf Proc IEEE Eng Med Biol Soc; 2007. pp. 6613–6616. [PubMed]
- Musallam S, Corneil BD, Greger B, Scherberger H, Andersen RA. Cognitive control signals for neural prosthetics. Science. 2004;305:258–262. [PubMed]
- Paz R, Vaadia E. Specificity of sensorimotor learning and the neural code: neuronal representations in the primary motor cortex. J Physiol Paris. 2005;98:331–348. [PubMed]
- Pfister J-P, Toyoizumi T, Barber D, Gerstner W. Optimal Spike-Timing-Dependent Plasticity for Precise Action Potential Firing in Supervised Learning. Neural Computation. 2006;18:1318–1348. [PubMed]
- Reynolds JN, Wickens JR. Dopamine-dependent plasticity of corticostriatal synapses. Neural Networks. 2002;15:507–521. [PubMed]
- Rokni U, Richardson AG, Bizzi E, Seung HS. Motor learning with unstable neural representations. Neuron. 2007;54:653–666. [PubMed]
- Schwartz AB. Useful signals from motor cortex. J Physiology. 2007;579:581–601. [PubMed]
- Seung HS. Learning in Spiking Neural Networks by Reinforcement of Stochastic Synaptic Transmission. Neuron. 2003;40:1063–1073. [PubMed]
- Sober SJ, Wohlgemuth MJ, Brainard MS. Central contributions to acoustic variation in birdsong. J Neuroscience. 2008;28:10370–10379. [PMC free article] [PubMed]
- Taylor DM, Tillery SI, Schwartz AB. Direct cortical control of 3D neuroprosthetic devices. Science. 2002;296:1829–1832. [PubMed]
- Tumer EC, Brainard MS. Performance variability enables adaptive plasticity of `crystallized' adult birdsong. Nature. 2007;250:1240–1244. [PubMed]
- Velliste M, Perel S, Spalding MC, Whitford AS, Schwartz AB. Cortical control of a prosthetic arm for self-feeding. Nature. 2008;453:1098–1101. [PubMed]
- Williams RJ. Simple Statistical Gradient-Following Algorithms for Connectionist Reinforcement Learning. Machine Learning. 1992;8:229–256.
- Xie X, Seung HS. Learning in neural networks by reinforcement of irregular spiking. Physical Review E. 2004;69 [PubMed]

PubMed Central Canada is a service of the Canadian Institutes of Health Research (CIHR) working in partnership with the National Research Council's national science library in cooperation with the National Center for Biotechnology Information at the U.S. National Library of Medicine(NCBI/NLM). It includes content provided to the PubMed Central International archive by participating publishers. |