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At this point you should ask yourself: "But how often do I need to draw the receiver or output without also drawing a neuron?". If the answer to that question is "probably never!", then you should just fold all of that functionality into the Neuron class:

import math
import random
import pygame


class Neuron:
    """A single neuron with Synapses that connect to others"""

    def __init__(self, x, y):
        self.x = x
        self.y = y
        self.pos = x, y
        self.width = 30
        self.offset = self.width / 2
        self.connections = []
        self.angle = math.radians(random.randint(0, 360))
        self.rect = (self.x - self.offset, self.y - self.offset,
                     self.width, self.width)
        self.color = 200, 150, 50

    def draw(self, surf):
        pygame.draw.ellipse(surf, self.color, self.rect)
        pygame.draw.ellipse(surf, *self.input)
        pygame.draw.ellipse(surf, *self.output)

    @property
    def input(self):
        color = 200, 255, 200
        rect = self.calc_rect()
        return color, rect

    @property
    def output(self):
        color = 200, 255, 200
        rect = self.calc_rect(output=True)
        return color, rect

    def calc_rect(self, output=False):
        width = 0.5 * self.width
        distance = 0.4 * self.width
        offset = 0.25 * self.width
        if output:
            pos = (self.x + math.sin(self.angle) * distance,
                   self.y - math.cos(self.angle) * distance)
        else:
            pos = (self.x - math.sin(self.angle) * distance,
                   self.y + math.cos(self.angle) * distance)
        return pos[0] - offset, pos[1] - offset, width, width

Here I did not go through the effort of making sure that Neuron.input and Neuron.output are only calculated once, but you could do it in the same way as I did above.

Note that I made all of your lists into tuples (like Neuron.pos). This is just aesthetics atm, but if you want to be able to do e.g. neuron.pos[0] += 1, then you need to make them lists again.

class NeuronIO:
    """Draw input or output of a neuron.

    Abstract base class.
    """
    def __init__(self, neuron):
        self.neuron = neuron

        width = 0.5 * self.neuron.width
        offset = width / 2
        distance = 0.4 * self.neuron.width
        self._pos = None
        self._rect = None
     
    def draw(self, surf):
        pygame.draw.ellipse(surf, [200, 150, 200], self.rect)

    @property
    def pos(self):
        """Needs to be implemented by the child classes"""
        raise NotImplementedError

    @property
    def rect(self):
        if self._rect is None:
            pos = self.pos
            self._rect = pos[0] - offset, pos[1] - offset, width, width 
        return self._rect


class NeuronOutput(NeuronIO):
    """The place where a Neuron outputs its signal"""
    @property
    def pos(self):
        if self._pos is None:
            self._pos = (self.neuron.x + math.sin(self.neuron.angle) * distance,
                         self.neuron.y - math.cos(self.neuron.angle) * distance)
        return self._pos


class NeuronReciever(NeuronIO):
    """Where a Neuron receives input"""
    @property
    def pos(self):
        if self._pos is None:
            self._pos = (self.neuron.x - math.sin(self.neuron.angle) * distance,
                         self.neuron.y + math.cos(self.neuron.angle) * distance)
        return self._pos

class NeuronIO:
    """Draw input or output of a neuron.

    Abstract base class.
    """
    color = 0, 0, 0  # default color is black

    def __init__(self, neuron):
        self.neuron = neuron

        width = 0.5 * self.neuron.width
        offset = width / 2
        distance = 0.4 * self.neuron.width
        self._pos = None
        self._rect = None
     
    def draw(self, surf):
        pygame.draw.ellipse(surf, self.color, self.rect)

    @property
    def pos(self):
        """Needs to be implemented by the child classes"""
        raise NotImplementedError

    @property
    def rect(self):
        if self._rect is None:
            pos = self.pos
            self._rect = pos[0] - offset, pos[1] - offset, width, width 
        return self._rect


class NeuronOutput(NeuronIO):
    """The place where a Neuron outputs its signal"""
    color = 200, 150, 200

    @property
    def pos(self):
        if self._pos is None:
            self._pos = (self.neuron.x + math.sin(self.neuron.angle) * distance,
                         self.neuron.y - math.cos(self.neuron.angle) * distance)
        return self._pos


class NeuronReciever(NeuronIO):
    """Where a Neuron receives input"""
    color = 200, 255, 200
    @property
    def pos(self):
        if self._pos is None:
            self._pos = (self.neuron.x - math.sin(self.neuron.angle) * distance,
                         self.neuron.y + math.cos(self.neuron.angle) * distance)
        return self._pos

class NeuronIO:
    
    def __init__(self, neuron):
        self.neuron = neuron

        width = 0.5 * self.neuron.width
        offset = width / 2
        distance = 0.4 * self.neuron.width
        self._pos = None
        self._rect = None
     
    def draw(self,surf):
        pygame.draw.ellipse(surf, [200, 150, 200], self.rect)

    @property
    def pos(self):
        """Needs to be implemented by the child classes"""
        raise NotImplementedError

    @property
    def rect(self):
        if self._rect is None:
            pos = self.pos
            self._rect = pos[0] - offset, pos[1] - offset, width, width 
        return self._rect

class NeuronOutput(NeuronIO):
    """The place where a Neuron outputs its signal"""
    @property
    def pos(self):
        if self._pos is None:
            self._pos = (self.neuron.x + math.sin(self.neuron.angle) * distance,
                         self.neuron.y - math.cos(self.neuron.angle) * distance)
        return self._pos


class NeuronReciever(NeuronIO):
    """Where a Neuron receives input"""
    @property
    def pos(self):
        if self._pos is None:
            self._pos = (self.neuron.x - math.sin(self.neuron.angle) * distance,
                         self.neuron.y + math.cos(self.neuron.angle) * distance)
        return self._pos

class NeuronIO:
    """Draw input or output of a neuron.

    Abstract base class.
    """
    def __init__(self, neuron):
        self.neuron = neuron

        width = 0.5 * self.neuron.width
        offset = width / 2
        distance = 0.4 * self.neuron.width
        self._pos = None
        self._rect = None
     
    def draw(self, surf):
        pygame.draw.ellipse(surf, [200, 150, 200], self.rect)

    @property
    def pos(self):
        """Needs to be implemented by the child classes"""
        raise NotImplementedError

    @property
    def rect(self):
        if self._rect is None:
            pos = self.pos
            self._rect = pos[0] - offset, pos[1] - offset, width, width 
        return self._rect 


class NeuronOutput(NeuronIO):
    """The place where a Neuron outputs its signal"""
    @property
    def pos(self):
        if self._pos is None:
            self._pos = (self.neuron.x + math.sin(self.neuron.angle) * distance,
                         self.neuron.y - math.cos(self.neuron.angle) * distance)
        return self._pos


class NeuronReciever(NeuronIO):
    """Where a Neuron receives input"""
    @property
    def pos(self):
        if self._pos is None:
            self._pos = (self.neuron.x - math.sin(self.neuron.angle) * distance,
                         self.neuron.y + math.cos(self.neuron.angle) * distance)
        return self._pos