# brainpy.neurons.FractionalIzhikevich#

class brainpy.neurons.FractionalIzhikevich(size, alpha, num_memory, a=0.02, b=0.2, c=-65.0, d=8.0, f=0.04, g=5.0, h=140.0, R=1.0, tau=1.0, V_th=30.0, V_initializer=OneInit(value=-65.0), u_initializer=OneInit(value=-13.0), keep_size=False, input_var=True, name=None)[source]#

Fractional-order Izhikevich model .

The fractional-order Izhikevich model is given by

\begin{split}\begin{aligned} &\tau \frac{d^{\alpha} v}{d t^{\alpha}}=\mathrm{f} v^{2}+g v+h-u+R I \\ &\tau \frac{d^{\alpha} u}{d t^{\alpha}}=a(b v-u) \end{aligned}\end{split}

where $$\alpha$$ is the fractional order (exponent) such that $$0<\alpha\le1$$. It is a commensurate system that reduces to classical Izhikevich model at $$\alpha=1$$.

The time $$t$$ is in ms; and the system variable $$v$$ expressed in mV corresponds to membrane voltage. Moreover, $$u$$ expressed in mV is the recovery variable that corresponds to the activation of K+ ionic current and inactivation of Na+ ionic current.

The parameters $$f, g, h$$ are fixed constants (should not be changed) such that $$f=0.04$$ (mV)−1, $$g=5, h=140$$ mV; and $$a$$ and $$b$$ are dimensionless parameters. The time constant $$\tau=1$$ ms; the resistance $$R=1$$ Ω; and $$I$$ expressed in mA measures the injected (applied) dc stimulus current to the system.

When the membrane voltage reaches the spike peak $$v_{peak}$$, the two variables are rest as follow:

$\begin{split}\text { if } v \geq v_{\text {peak }} \text { then }\left\{\begin{array}{l} v \leftarrow c \\ u \leftarrow u+d \end{array}\right.\end{split}$

we used $$v_{peak}=30$$ mV, and $$c$$ and $$d$$ are parameters expressed in mV. When the spike reaches its peak value, the membrane voltage $$v$$ and the recovery variable $$u$$ are reset according to the above condition.

Examples

References

__init__(size, alpha, num_memory, a=0.02, b=0.2, c=-65.0, d=8.0, f=0.04, g=5.0, h=140.0, R=1.0, tau=1.0, V_th=30.0, V_initializer=OneInit(value=-65.0), u_initializer=OneInit(value=-13.0), keep_size=False, input_var=True, name=None)[source]#

Methods

 __init__(size, alpha, num_memory[, a, b, c, ...]) add_aft_update(key, fun) Add the after update into this node add_bef_update(key, fun) Add the before update into this node add_inp_fun(key, fun) Add an input function. clear_input() Empty function of clearing inputs. cpu() Move all variable into the CPU device. cuda() Move all variables into the GPU device. dV(V, t, u, I_ext) du(u, t, V) get_aft_update(key) Get the after update of this node by the given key. get_batch_shape([batch_size]) get_bef_update(key) Get the before update of this node by the given key. get_delay_data(identifier, delay_pos, *indices) Get delay data according to the provided delay steps. get_delay_var(name) get_inp_fun(key) Get the input function. get_local_delay(var_name, delay_name) Get the delay at the given identifier (name). has_aft_update(key) Whether this node has the after update of the given key. has_bef_update(key) Whether this node has the before update of the given key. init_param(param[, shape, sharding]) Initialize parameters. init_variable(var_data, batch_or_mode[, ...]) Initialize variables. jit_step_run(i, *args, **kwargs) The jitted step function for running. load_state(state_dict, **kwargs) Load states from a dictionary. load_state_dict(state_dict[, warn, compatible]) Copy parameters and buffers from state_dict into this module and its descendants. nodes([method, level, include_self]) Collect all children nodes. register_delay(identifier, delay_step, ...) Register delay variable. register_implicit_nodes(*nodes[, node_cls]) register_implicit_vars(*variables[, var_cls]) register_local_delay(var_name, delay_name[, ...]) Register local relay at the given delay time. reset(*args, **kwargs) Reset function which reset the whole variables in the model (including its children models). reset_local_delays([nodes]) Reset local delay variables. reset_state([batch_size]) Reset function which resets local states in this model. return_info() rtype: Union[Variable, ReturnInfo] save_state(**kwargs) Save states as a dictionary. setattr(key, value) rtype: None state_dict(**kwargs) Returns a dictionary containing a whole state of the module. step_run(i, *args, **kwargs) The step run function. sum_inputs(*args[, init, label]) Summarize all inputs by the defined input functions .cur_inputs. to(device) Moves all variables into the given device. tpu() Move all variables into the TPU device. tracing_variable(name, init, shape[, ...]) Initialize the variable which can be traced during computations and transformations. train_vars([method, level, include_self]) The shortcut for retrieving all trainable variables. tree_flatten() Flattens the object as a PyTree. tree_unflatten(aux, dynamic_values) Unflatten the data to construct an object of this class. unique_name([name, type_]) Get the unique name for this object. update([x]) The function to specify the updating rule. update_local_delays([nodes]) Update local delay variables. vars([method, level, include_self, ...]) Collect all variables in this node and the children nodes.

Attributes

 derivative mode Mode of the model, which is useful to control the multiple behaviors of the model. name Name of the model. supported_modes Supported computing modes. varshape The shape of variables in the neuron group. cur_inputs