BindsNET学习系列 —— LearningRule

相关源码：bindsnet/bindsnet/learning/learning.py

1、LearningRule

class LearningRule(ABC):
    # language=rst
    """
    Abstract base class for learning rules.
    """

    def __init__(
        self,
        connection: AbstractConnection,
        nu: Optional[Union[float, Sequence[float]]] = None,
        reduction: Optional[callable] = None,
        weight_decay: float = 0.0,
        **kwargs
    ) -> None:
        # language=rst
        """
        Abstract constructor for the ``LearningRule`` object.

        :param connection: An ``AbstractConnection`` object.
        :param nu: Single or pair of learning rates for pre- and post-synaptic events.
        :param reduction: Method for reducing parameter updates along the batch
            dimension.
        :param weight_decay: Constant multiple to decay weights by on each iteration.
        """
        # Connection parameters.
        self.connection = connection
        self.source = connection.source
        self.target = connection.target

        self.wmin = connection.wmin
        self.wmax = connection.wmax

        # Learning rate(s).
        if nu is None:
            nu = [0.0, 0.0]
        elif isinstance(nu, float) or isinstance(nu, int):
            nu = [nu, nu]

        self.nu = torch.zeros(2, dtype=torch.float)
        self.nu[0] = nu[0]
        self.nu[1] = nu[1]

        # Parameter update reduction across minibatch dimension.
        if reduction is None:
            if self.source.batch_size == 1:
                self.reduction = torch.squeeze
            else:
                self.reduction = torch.sum
        else:
            self.reduction = reduction

        # Weight decay.
        self.weight_decay = 1.0 - weight_decay if weight_decay else 1.0

    def update(self) -> None:
        # language=rst
        """
        Abstract method for a learning rule update.
        """
        # Implement weight decay.
        if self.weight_decay:
            self.connection.w *= self.weight_decay

        # Bound weights.
        if (
            self.connection.wmin != -np.inf or self.connection.wmax != np.inf
        ) and not isinstance(self, NoOp):
            self.connection.w.clamp_(self.connection.wmin, self.connection.wmax)

学习规则的抽象基类，可以包括权重衰减和权重截断。

2、NoOp（没有效果的学习规则）

class NoOp(LearningRule):
    # language=rst
    """
    Learning rule with no effect.
    """

    def __init__(
        self,
        connection: AbstractConnection,
        nu: Optional[Union[float, Sequence[float]]] = None,
        reduction: Optional[callable] = None,
        weight_decay: float = 0.0,
        **kwargs
    ) -> None:
        # language=rst
        """
        Abstract constructor for the ``LearningRule`` object.

        :param connection: An ``AbstractConnection`` object.
        :param nu: Single or pair of learning rates for pre- and post-synaptic events.
        :param reduction: Method for reducing parameter updates along the batch
            dimension.
        :param weight_decay: Constant multiple to decay weights by on each iteration.
        """
        super().__init__(
            connection=connection,
            nu=nu,
            reduction=reduction,
            weight_decay=weight_decay,
            **kwargs
        )

    def update(self, **kwargs) -> None:
        # language=rst
        """
        Abstract method for a learning rule update.
        """
        super().update()

没有效果的学习规则(默认)，直接继承自学习规则的抽象基类。

3、PostPre（STDP的在线实现）

class PostPre(LearningRule):
    # language=rst
    """
    Simple STDP rule involving both pre- and post-synaptic spiking activity. By default,
    pre-synaptic update is negative and the post-synaptic update is positive.
    """

    def __init__(
        self,
        connection: AbstractConnection,
        nu: Optional[Union[float, Sequence[float]]] = None,
        reduction: Optional[callable] = None,
        weight_decay: float = 0.0,
        **kwargs
    ) -> None:
        # language=rst
        """
        Constructor for ``PostPre`` learning rule.

        :param connection: An ``AbstractConnection`` object whose weights the
            ``PostPre`` learning rule will modify.
        :param nu: Single or pair of learning rates for pre- and post-synaptic events.
        :param reduction: Method for reducing parameter updates along the batch
            dimension.
        :param weight_decay: Constant multiple to decay weights by on each iteration.
        """
        super().__init__(
            connection=connection,
            nu=nu,
            reduction=reduction,
            weight_decay=weight_decay,
            **kwargs
        )

        assert (
            self.source.traces and self.target.traces
        ), "Both pre- and post-synaptic nodes must record spike traces."

        if isinstance(connection, (Connection, LocalConnection)):
            self.update = self._connection_update
        elif isinstance(connection, Conv2dConnection):
            self.update = self._conv2d_connection_update
        else:
            raise NotImplementedError(
                "This learning rule is not supported for this Connection type."
            )

    def _connection_update(self, **kwargs) -> None:
        # language=rst
        """
        Post-pre learning rule for ``Connection`` subclass of ``AbstractConnection``
        class.
        """
        batch_size = self.source.batch_size

        # Pre-synaptic update. 突触前脉冲发放时基于突触后迹更新突触权重（下降）
        if self.nu[0]:
            source_s = self.source.s.view(batch_size, -1).unsqueeze(2).float()
            target_x = self.target.x.view(batch_size, -1).unsqueeze(1) * self.nu[0]
            self.connection.w -= self.reduction(torch.bmm(source_s, target_x), dim=0)
            del source_s, target_x

        # Post-synaptic update. 突触后脉冲发放时基于突触前迹更新突触权重（上升）
        if self.nu[1]:
            target_s = self.target.s.view(batch_size, -1).unsqueeze(1).float() * self.nu[1]
            source_x = self.source.x.view(batch_size, -1).unsqueeze(2)
            self.connection.w += self.reduction(torch.bmm(source_x, target_s), dim=0)
            del source_x, target_s

        super().update()

    def _conv2d_connection_update(self, **kwargs) -> None:
        # language=rst
        """
        Post-pre learning rule for ``Conv2dConnection`` subclass of
        ``AbstractConnection`` class.
        """
        # Get convolutional layer parameters.
        out_channels, _, kernel_height, kernel_width = self.connection.w.size()
        padding, stride = self.connection.padding, self.connection.stride
        batch_size = self.source.batch_size

        # Reshaping spike traces and spike occurrences.
        source_x = im2col_indices(
            self.source.x, kernel_height, kernel_width, padding=padding, stride=stride
        )
        target_x = self.target.x.view(batch_size, out_channels, -1)
        source_s = im2col_indices(
            self.source.s.float(),
            kernel_height,
            kernel_width,
            padding=padding,
            stride=stride,
        )
        target_s = self.target.s.view(batch_size, out_channels, -1).float()

        # Pre-synaptic update.
        if self.nu[0]:
            pre = self.reduction(
                torch.bmm(target_x, source_s.permute((0, 2, 1))), dim=0
            )
            self.connection.w -= self.nu[0] * pre.view(self.connection.w.size())

        # Post-synaptic update.
        if self.nu[1]:
            post = self.reduction(
                torch.bmm(target_s, source_x.permute((0, 2, 1))), dim=0
            )
            self.connection.w += self.nu[1] * post.view(self.connection.w.size())

        super().update()

简单的STDP规则，包括突触前和突触后的脉冲活动。默认情况下，突触前更新为负(-学习率₁ × 突触前脉冲 × 突触后发放迹)，突触后更新为正(学习率₂ × 突触前发放迹 × 突触后脉冲)。

4、WeightDependentPostPre

class WeightDependentPostPre(LearningRule):
    # language=rst
    """
    STDP rule involving both pre- and post-synaptic spiking activity. The post-synaptic
    update is positive and the pre- synaptic update is negative, and both are dependent
    on the magnitude of the synaptic weights.
    """

    def __init__(
        self,
        connection: AbstractConnection,
        nu: Optional[Union[float, Sequence[float]]] = None,
        reduction: Optional[callable] = None,
        weight_decay: float = 0.0,
        **kwargs
    ) -> None:
        # language=rst
        """
        Constructor for ``WeightDependentPostPre`` learning rule.

        :param connection: An ``AbstractConnection`` object whose weights the
            ``WeightDependentPostPre`` learning rule will modify.
        :param nu: Single or pair of learning rates for pre- and post-synaptic events.
        :param reduction: Method for reducing parameter updates along the batch
            dimension.
        :param weight_decay: Constant multiple to decay weights by on each iteration.
        """
        super().__init__(
            connection=connection,
            nu=nu,
            reduction=reduction,
            weight_decay=weight_decay,
            **kwargs
        )

        assert self.source.traces, "Pre-synaptic nodes must record spike traces."
        assert (
            connection.wmin != -np.inf and connection.wmax != np.inf
        ), "Connection must define finite wmin and wmax."

        self.wmin = connection.wmin
        self.wmax = connection.wmax

        if isinstance(connection, (Connection, LocalConnection)):
            self.update = self._connection_update
        elif isinstance(connection, Conv2dConnection):
            self.update = self._conv2d_connection_update
        else:
            raise NotImplementedError(
                "This learning rule is not supported for this Connection type."
            )

    def _connection_update(self, **kwargs) -> None:
        # language=rst
        """
        Post-pre learning rule for ``Connection`` subclass of ``AbstractConnection``
        class.
        """
        batch_size = self.source.batch_size

        source_s = self.source.s.view(batch_size, -1).unsqueeze(2).float()
        source_x = self.source.x.view(batch_size, -1).unsqueeze(2)
        target_s = self.target.s.view(batch_size, -1).unsqueeze(1).float()
        target_x = self.target.x.view(batch_size, -1).unsqueeze(1)

        update = 0

        # Pre-synaptic update.
        if self.nu[0]:
            outer_product = self.reduction(torch.bmm(source_s, target_x), dim=0)
            update -= self.nu[0] * outer_product * (self.connection.w - self.wmin)

        # Post-synaptic update.
        if self.nu[1]:
            outer_product = self.reduction(torch.bmm(source_x, target_s), dim=0)
            update += self.nu[1] * outer_product * (self.wmax - self.connection.w)

        self.connection.w += update

        super().update()

    def _conv2d_connection_update(self, **kwargs) -> None:
        # language=rst
        """
        Post-pre learning rule for ``Conv2dConnection`` subclass of
        ``AbstractConnection`` class.
        """
        # Get convolutional layer parameters.
        (
            out_channels,
            in_channels,
            kernel_height,
            kernel_width,
        ) = self.connection.w.size()
        padding, stride = self.connection.padding, self.connection.stride
        batch_size = self.source.batch_size

        # Reshaping spike traces and spike occurrences.
        source_x = im2col_indices(
            self.source.x, kernel_height, kernel_width, padding=padding, stride=stride
        )
        target_x = self.target.x.view(batch_size, out_channels, -1)
        source_s = im2col_indices(
            self.source.s.float(),
            kernel_height,
            kernel_width,
            padding=padding,
            stride=stride,
        )
        target_s = self.target.s.view(batch_size, out_channels, -1).float()

        update = 0

        # Pre-synaptic update.
        if self.nu[0]:
            pre = self.reduction(
                torch.bmm(target_x, source_s.permute((0, 2, 1))), dim=0
            )
            update -= (
                self.nu[0]
                * pre.view(self.connection.w.size())
                * (self.connection.w - self.wmin)
            )

        # Post-synaptic update.
        if self.nu[1]:
            post = self.reduction(
                torch.bmm(target_s, source_x.permute((0, 2, 1))), dim=0
            )
            update += (
                self.nu[1]
                * post.view(self.connection.w.size())
                * (self.wmax - self.connection.wmin)
            )

        self.connection.w += update

        super().update()

STDP规则涉及突触前和突触后的脉冲活动。突触前更新为负(-学习率₁ × 突触前脉冲 × 突触后发放迹 × (weight - weight_min))，突触后更新为正(学习率₂ × 突触前发放迹 × 突触后脉冲 × (weight_max - weight))，两者都依赖于突触权重的大小。

5、Hebbian

class Hebbian(LearningRule):
    # language=rst
    """
    Simple Hebbian learning rule. Pre- and post-synaptic updates are both positive.
    """

    def __init__(
        self,
        connection: AbstractConnection,
        nu: Optional[Union[float, Sequence[float]]] = None,
        reduction: Optional[callable] = None,
        weight_decay: float = 0.0,
        **kwargs
    ) -> None:
        # language=rst
        """
        Constructor for ``Hebbian`` learning rule.

        :param connection: An ``AbstractConnection`` object whose weights the
            ``Hebbian`` learning rule will modify.
        :param nu: Single or pair of learning rates for pre- and post-synaptic events.
        :param reduction: Method for reducing parameter updates along the batch
            dimension.
        :param weight_decay: Constant multiple to decay weights by on each iteration.
        """
        super().__init__(
            connection=connection,
            nu=nu,
            reduction=reduction,
            weight_decay=weight_decay,
            **kwargs
        )

        assert (
            self.source.traces and self.target.traces
        ), "Both pre- and post-synaptic nodes must record spike traces."

        if isinstance(connection, (Connection, LocalConnection)):
            self.update = self._connection_update
        elif isinstance(connection, Conv2dConnection):
            self.update = self._conv2d_connection_update
        else:
            raise NotImplementedError(
                "This learning rule is not supported for this Connection type."
            )

    def _connection_update(self, **kwargs) -> None:
        # language=rst
        """
        Hebbian learning rule for ``Connection`` subclass of ``AbstractConnection``
        class.
        """
        batch_size = self.source.batch_size

        source_s = self.source.s.view(batch_size, -1).unsqueeze(2).float()
        source_x = self.source.x.view(batch_size, -1).unsqueeze(2)
        target_s = self.target.s.view(batch_size, -1).unsqueeze(1).float()
        target_x = self.target.x.view(batch_size, -1).unsqueeze(1)

        # Pre-synaptic update.
        update = self.reduction(torch.bmm(source_s, target_x), dim=0)
        self.connection.w += self.nu[0] * update

        # Post-synaptic update.
        update = self.reduction(torch.bmm(source_x, target_s), dim=0)
        self.connection.w += self.nu[1] * update

        super().update()

    def _conv2d_connection_update(self, **kwargs) -> None:
        # language=rst
        """
        Hebbian learning rule for ``Conv2dConnection`` subclass of
        ``AbstractConnection`` class.
        """
        out_channels, _, kernel_height, kernel_width = self.connection.w.size()
        padding, stride = self.connection.padding, self.connection.stride
        batch_size = self.source.batch_size

        # Reshaping spike traces and spike occurrences.
        source_x = im2col_indices(
            self.source.x, kernel_height, kernel_width, padding=padding, stride=stride
        )
        target_x = self.target.x.view(batch_size, out_channels, -1)
        source_s = im2col_indices(
            self.source.s.float(),
            kernel_height,
            kernel_width,
            padding=padding,
            stride=stride,
        )
        target_s = self.target.s.view(batch_size, out_channels, -1).float()

        # Pre-synaptic update.
        pre = self.reduction(torch.bmm(target_x, source_s.permute((0, 2, 1))), dim=0)
        self.connection.w += self.nu[0] * pre.view(self.connection.w.size())

        # Post-synaptic update.
        post = self.reduction(torch.bmm(target_s, source_x.permute((0, 2, 1))), dim=0)
        self.connection.w += self.nu[1] * post.view(self.connection.w.size())

        super().update()

简单的Hebbian学习规则。突触前后的更新都为正(分别为学习率₁ × 突触前脉冲 × 突触后发放迹，以及学习率₂ × 突触前发放迹× 突触后脉冲)。

6、MSTDP

class MSTDP(LearningRule):
    # language=rst
    """
    Reward-modulated STDP. Adapted from `(Florian 2007)
    <https://florian.io/papers/2007_Florian_Modulated_STDP.pdf>`_.
    """

    def __init__(
        self,
        connection: AbstractConnection,
        nu: Optional[Union[float, Sequence[float]]] = None,
        reduction: Optional[callable] = None,
        weight_decay: float = 0.0,
        **kwargs
    ) -> None:
        # language=rst
        """
        Constructor for ``MSTDP`` learning rule.

        :param connection: An ``AbstractConnection`` object whose weights the ``MSTDP``
            learning rule will modify.
        :param nu: Single or pair of learning rates for pre- and post-synaptic events,
            respectively.
        :param reduction: Method for reducing parameter updates along the minibatch
            dimension.
        :param weight_decay: Constant multiple to decay weights by on each iteration.

        Keyword arguments:

        :param tc_plus: Time constant for pre-synaptic firing trace.
        :param tc_minus: Time constant for post-synaptic firing trace.
        """
        super().__init__(
            connection=connection,
            nu=nu,
            reduction=reduction,
            weight_decay=weight_decay,
            **kwargs
        )

        if isinstance(connection, (Connection, LocalConnection)):
            self.update = self._connection_update
        elif isinstance(connection, Conv2dConnection):
            self.update = self._conv2d_connection_update
        else:
            raise NotImplementedError(
                "This learning rule is not supported for this Connection type."
            )

        self.tc_plus = torch.tensor(kwargs.get("tc_plus", 20.0))
        self.tc_minus = torch.tensor(kwargs.get("tc_minus", 20.0))

    def _connection_update(self, **kwargs) -> None:
        # language=rst
        """
        MSTDP learning rule for ``Connection`` subclass of ``AbstractConnection`` class.

        Keyword arguments:

        :param Union[float, torch.Tensor] reward: Reward signal from reinforcement
            learning task.
        :param float a_plus: Learning rate (post-synaptic).
        :param float a_minus: Learning rate (pre-synaptic).
        """
        batch_size = self.source.batch_size

        # Initialize eligibility, P^+, and P^-.
        if not hasattr(self, "p_plus"):
            self.p_plus = torch.zeros(
                batch_size, *self.source.shape, device=self.source.s.device
            )
        if not hasattr(self, "p_minus"):
            self.p_minus = torch.zeros(
                batch_size, *self.target.shape, device=self.target.s.device
            )
        if not hasattr(self, "eligibility"):
            self.eligibility = torch.zeros(
                batch_size, *self.connection.w.shape, device=self.connection.w.device
            )

        # Reshape pre- and post-synaptic spikes.
        source_s = self.source.s.view(batch_size, -1).float()
        target_s = self.target.s.view(batch_size, -1).float()

        # Parse keyword arguments.
        reward = kwargs["reward"]
        a_plus = torch.tensor(
            kwargs.get("a_plus", 1.0), device=self.connection.w.device
        )
        a_minus = torch.tensor(
            kwargs.get("a_minus", -1.0), device=self.connection.w.device
        )

        # Compute weight update based on the eligibility value of the past timestep.
        update = reward * self.eligibility
        self.connection.w += self.nu[0] * self.reduction(update, dim=0)

        # Update P^+ and P^- values.
        self.p_plus *= torch.exp(-self.connection.dt / self.tc_plus)
        self.p_plus += a_plus * source_s
        self.p_minus *= torch.exp(-self.connection.dt / self.tc_minus)
        self.p_minus += a_minus * target_s

        # Calculate point eligibility value.
        self.eligibility = torch.bmm(
            self.p_plus.unsqueeze(2), target_s.unsqueeze(1)
        ) + torch.bmm(source_s.unsqueeze(2), self.p_minus.unsqueeze(1))

        super().update()

    def _conv2d_connection_update(self, **kwargs) -> None:
        # language=rst
        """
        MSTDP learning rule for ``Conv2dConnection`` subclass of ``AbstractConnection``
        class.

        Keyword arguments:

        :param Union[float, torch.Tensor] reward: Reward signal from reinforcement
            learning task.
        :param float a_plus: Learning rate (post-synaptic).
        :param float a_minus: Learning rate (pre-synaptic).
        """
        batch_size = self.source.batch_size

        # Initialize eligibility.
        if not hasattr(self, "eligibility"):
            self.eligibility = torch.zeros(
                batch_size, *self.connection.w.shape, device=self.connection.w.device
            )

        # Parse keyword arguments.
        reward = kwargs["reward"]
        a_plus = torch.tensor(
            kwargs.get("a_plus", 1.0), device=self.connection.w.device
        )
        a_minus = torch.tensor(
            kwargs.get("a_minus", -1.0), device=self.connection.w.device
        )

        batch_size = self.source.batch_size

        # Compute weight update based on the eligibility value of the past timestep.
        update = reward * self.eligibility
        self.connection.w += self.nu[0] * torch.sum(update, dim=0)

        out_channels, _, kernel_height, kernel_width = self.connection.w.size()
        padding, stride = self.connection.padding, self.connection.stride

        # Initialize P^+ and P^-.
        if not hasattr(self, "p_plus"):
            self.p_plus = torch.zeros(
                batch_size, *self.source.shape, device=self.connection.w.device
            )
            self.p_plus = im2col_indices(
                self.p_plus, kernel_height, kernel_width, padding=padding, stride=stride
            )
        if not hasattr(self, "p_minus"):
            self.p_minus = torch.zeros(
                batch_size, *self.target.shape, device=self.connection.w.device
            )
            self.p_minus = self.p_minus.view(batch_size, out_channels, -1).float()

        # Reshaping spike occurrences.
        source_s = im2col_indices(
            self.source.s.float(),
            kernel_height,
            kernel_width,
            padding=padding,
            stride=stride,
        )
        target_s = self.target.s.view(batch_size, out_channels, -1).float()

        # Update P^+ and P^- values. 前者跟踪突触前脉冲的影响，后者跟踪突触后脉冲的影响
        self.p_plus *= torch.exp(-self.connection.dt / self.tc_plus)
        self.p_plus += a_plus * source_s
        self.p_minus *= torch.exp(-self.connection.dt / self.tc_minus)
        self.p_minus += a_minus * target_s

        # Calculate point eligibility value.
        self.eligibility = torch.bmm(
            target_s, self.p_plus.permute((0, 2, 1))
        ) + torch.bmm(self.p_minus, source_s.permute((0, 2, 1)))
        self.eligibility = self.eligibility.view(self.connection.w.size())

        super().update()

奖励调节STDP (R-STDP)，改编自(Florian 2007)<https://florian.io/papers/2007_Florian_Modulated_STDP.pdf>

论文参见：Reinforcement Learning Through Modulation of Spike-Timing-Dependent Synaptic Plasticity - 穷酸秀才大艹包 - 博客园 (cnblogs.com)

7、MSTDPET

class MSTDPET(LearningRule):
    # language=rst
    """
    Reward-modulated STDP with eligibility trace. Adapted from
    `(Florian 2007) <https://florian.io/papers/2007_Florian_Modulated_STDP.pdf>`_.
    """

    def __init__(
        self,
        connection: AbstractConnection,
        nu: Optional[Union[float, Sequence[float]]] = None,
        reduction: Optional[callable] = None,
        weight_decay: float = 0.0,
        **kwargs
    ) -> None:
        # language=rst
        """
        Constructor for ``MSTDPET`` learning rule.

        :param connection: An ``AbstractConnection`` object whose weights the
            ``MSTDPET`` learning rule will modify.
        :param nu: Single or pair of learning rates for pre- and post-synaptic events,
            respectively.
        :param reduction: Method for reducing parameter updates along the minibatch
            dimension.
        :param weight_decay: Constant multiple to decay weights by on each iteration.

        Keyword arguments:

        :param float tc_plus: Time constant for pre-synaptic firing trace.
        :param float tc_minus: Time constant for post-synaptic firing trace.
        :param float tc_e_trace: Time constant for the eligibility trace.
        """
        super().__init__(
            connection=connection,
            nu=nu,
            reduction=reduction,
            weight_decay=weight_decay,
            **kwargs
        )

        if isinstance(connection, (Connection, LocalConnection)):
            self.update = self._connection_update
        elif isinstance(connection, Conv2dConnection):
            self.update = self._conv2d_connection_update
        else:
            raise NotImplementedError(
                "This learning rule is not supported for this Connection type."
            )

        self.tc_plus = torch.tensor(kwargs.get("tc_plus", 20.0))
        self.tc_minus = torch.tensor(kwargs.get("tc_minus", 20.0))
        self.tc_e_trace = torch.tensor(kwargs.get("tc_e_trace", 25.0))

    def _connection_update(self, **kwargs) -> None:
        # language=rst
        """
        MSTDPET learning rule for ``Connection`` subclass of ``AbstractConnection``
        class.

        Keyword arguments:

        :param Union[float, torch.Tensor] reward: Reward signal from reinforcement
            learning task.
        :param float a_plus: Learning rate (post-synaptic).
        :param float a_minus: Learning rate (pre-synaptic).
        """
        # Initialize eligibility, eligibility trace, P^+, and P^-.
        if not hasattr(self, "p_plus"):
            self.p_plus = torch.zeros((self.source.n), device=self.source.s.device)
        if not hasattr(self, "p_minus"):
            self.p_minus = torch.zeros((self.target.n), device=self.target.s.device)
        if not hasattr(self, "eligibility"):
            self.eligibility = torch.zeros(
                *self.connection.w.shape, device=self.connection.w.device
            )
        if not hasattr(self, "eligibility_trace"):
            self.eligibility_trace = torch.zeros(
                *self.connection.w.shape, device=self.connection.w.device
            )

        # Reshape pre- and post-synaptic spikes.
        source_s = self.source.s.view(-1).float()
        target_s = self.target.s.view(-1).float()

        # Parse keyword arguments.
        reward = kwargs["reward"]
        a_plus = torch.tensor(
            kwargs.get("a_plus", 1.0), device=self.connection.w.device
        )
        a_minus = torch.tensor(
            kwargs.get("a_minus", -1.0), device=self.connection.w.device
        )

        # Calculate value of eligibility trace based on the value
        # of the point eligibility value of the past timestep.
        self.eligibility_trace *= torch.exp(-self.connection.dt / self.tc_e_trace)
        self.eligibility_trace += self.eligibility / self.tc_e_trace

        # Compute weight update.
        self.connection.w += (
            self.nu[0] * self.connection.dt * reward * self.eligibility_trace
        )

        # Update P^+ and P^- values.
        self.p_plus *= torch.exp(-self.connection.dt / self.tc_plus)
        self.p_plus += a_plus * source_s
        self.p_minus *= torch.exp(-self.connection.dt / self.tc_minus)
        self.p_minus += a_minus * target_s

        # Calculate point eligibility value.
        self.eligibility = torch.ger(self.p_plus, target_s) + torch.ger(
            source_s, self.p_minus
        )

        super().update()

    def _conv2d_connection_update(self, **kwargs) -> None:
        # language=rst
        """
        MSTDPET learning rule for ``Conv2dConnection`` subclass of
        ``AbstractConnection`` class.

        Keyword arguments:

        :param Union[float, torch.Tensor] reward: Reward signal from reinforcement
            learning task.
        :param float a_plus: Learning rate (post-synaptic).
        :param float a_minus: Learning rate (pre-synaptic).
        """
        batch_size = self.source.batch_size

        # Initialize eligibility and eligibility trace.
        if not hasattr(self, "eligibility"):
            self.eligibility = torch.zeros(
                batch_size, *self.connection.w.shape, device=self.connection.w.device
            )
        if not hasattr(self, "eligibility_trace"):
            self.eligibility_trace = torch.zeros(
                batch_size, *self.connection.w.shape, device=self.connection.w.device
            )

        # Parse keyword arguments.
        reward = kwargs["reward"]
        a_plus = torch.tensor(
            kwargs.get("a_plus", 1.0), device=self.connection.w.device
        )
        a_minus = torch.tensor(
            kwargs.get("a_minus", -1.0), device=self.connection.w.device
        )

        # Calculate value of eligibility trace based on the value
        # of the point eligibility value of the past timestep.
        self.eligibility_trace *= torch.exp(-self.connection.dt / self.tc_e_trace)

        # Compute weight update.
        update = reward * self.eligibility_trace
        self.connection.w += self.nu[0] * self.connection.dt * torch.sum(update, dim=0)

        out_channels, _, kernel_height, kernel_width = self.connection.w.size()
        padding, stride = self.connection.padding, self.connection.stride

        # Initialize P^+ and P^-.
        if not hasattr(self, "p_plus"):
            self.p_plus = torch.zeros(
                batch_size, *self.source.shape, device=self.connection.w.device
            )
            self.p_plus = im2col_indices(
                self.p_plus, kernel_height, kernel_width, padding=padding, stride=stride
            )
        if not hasattr(self, "p_minus"):
            self.p_minus = torch.zeros(
                batch_size, *self.target.shape, device=self.connection.w.device
            )
            self.p_minus = self.p_minus.view(batch_size, out_channels, -1).float()

        # Reshaping spike occurrences.
        source_s = im2col_indices(
            self.source.s.float(),
            kernel_height,
            kernel_width,
            padding=padding,
            stride=stride,
        )
        target_s = (
            self.target.s.permute(1, 2, 3, 0).view(batch_size, out_channels, -1).float()
        )

        # Update P^+ and P^- values.
        self.p_plus *= torch.exp(-self.connection.dt / self.tc_plus)
        self.p_plus += a_plus * source_s
        self.p_minus *= torch.exp(-self.connection.dt / self.tc_minus)
        self.p_minus += a_minus * target_s

        # Calculate point eligibility value.
        self.eligibility = torch.bmm(
            target_s, self.p_plus.permute((0, 2, 1))
        ) + torch.bmm(self.p_minus, source_s.permute((0, 2, 1)))
        self.eligibility = self.eligibility.view(self.connection.w.size())

        super().update()

带资格迹的奖励调节STDP (R-STDP with eligibility trace)，改编自(Florian 2007)<https://florian.io/papers/2007_Florian_Modulated_STDP.pdf>

论文参见：Reinforcement Learning Through Modulation of Spike-Timing-Dependent Synaptic Plasticity - 穷酸秀才大艹包 - 博客园 (cnblogs.com)

8、Rmax

class Rmax(LearningRule):
    # language=rst
    """
    Reward-modulated learning rule derived from reward maximization principles. Adapted
    from `(Vasilaki et al., 2009)
    <https://intranet.physio.unibe.ch/Publikationen/Dokumente/Vasilaki2009PloSComputBio_1.pdf>`_.
    """

    def __init__(
        self,
        connection: AbstractConnection,
        nu: Optional[Union[float, Sequence[float]]] = None,
        reduction: Optional[callable] = None,
        weight_decay: float = 0.0,
        **kwargs
    ) -> None:
        # language=rst
        """
        Constructor for ``R-max`` learning rule.

        :param connection: An ``AbstractConnection`` object whose weights the ``R-max``
            learning rule will modify.
        :param nu: Single or pair of learning rates for pre- and post-synaptic events,
            respectively.
        :param reduction: Method for reducing parameter updates along the minibatch
            dimension.
        :param weight_decay: Constant multiple to decay weights by on each iteration.

        Keyword arguments:

        :param float tc_c: Time constant for balancing naive Hebbian and policy gradient
            learning.
        :param float tc_e_trace: Time constant for the eligibility trace.
        """
        super().__init__(
            connection=connection,
            nu=nu,
            reduction=reduction,
            weight_decay=weight_decay,
            **kwargs
        )

        # Trace is needed for computing epsilon.
        assert (
            self.source.traces and self.source.traces_additive
        ), "Pre-synaptic nodes must use additive spike traces."

        # Derivation of R-max depends on stochastic SRM neurons!
        assert isinstance(
            self.target, SRM0Nodes
        ), "R-max needs stochastically firing neurons, use SRM0Nodes."

        if isinstance(connection, (Connection, LocalConnection)):
            self.update = self._connection_update
        else:
            raise NotImplementedError(
                "This learning rule is not supported for this Connection type."
            )

        self.tc_c = torch.tensor(
            kwargs.get("tc_c", 5.0)
        )  # 0 for pure naive Hebbian, inf for pure policy gradient.
        self.tc_e_trace = torch.tensor(kwargs.get("tc_e_trace", 25.0))

    def _connection_update(self, **kwargs) -> None:
        # language=rst
        """
        R-max learning rule for ``Connection`` subclass of ``AbstractConnection`` class.

        Keyword arguments:

        :param Union[float, torch.Tensor] reward: Reward signal from reinforcement
            learning task.
        """
        # Initialize eligibility trace.
        if not hasattr(self, "eligibility_trace"):
            self.eligibility_trace = torch.zeros(
                *self.connection.w.shape, device=self.connection.w.device
            )

        # Reshape variables.
        target_s = self.target.s.view(-1).float()
        target_s_prob = self.target.s_prob.view(-1)
        source_x = self.source.x.view(-1)

        # Parse keyword arguments.
        reward = kwargs["reward"]

        # New eligibility trace.
        self.eligibility_trace *= 1 - self.connection.dt / self.tc_e_trace
        self.eligibility_trace += (
            target_s
            - (target_s_prob / (1.0 + self.tc_c / self.connection.dt * target_s_prob))
        ) * source_x[:, None]

        # Compute weight update.
        self.connection.w += self.nu[0] * reward * self.eligibility_trace

        super().update()

基于奖励最大化原理的奖励调节学习规则，改编自(Vasilaki et al., 2009)<https://intranet.physio.unibe.ch/Publikationen/Dokumente/Vasilaki2009PloSComputBio_1.pdf>